Page 1
C H A P T E R 9Topics in Analytic Geometry
Section 9.1 Circles and Parabolas . . . . . . . . . . . . . . . . . . . . 772
Section 9.2 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
Section 9.3 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . 795
Section 9.4 Rotation and Systems of Quadratic Equations . . . . . . . 807
Section 9.5 Parametric Equations . . . . . . . . . . . . . . . . . . . . 825
Section 9.6 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . 833
Section 9.7 Graphs of Polar Equations . . . . . . . . . . . . . . . . . 845
Section 9.8 Polar Equations of Conics . . . . . . . . . . . . . . . . . 854
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886
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Page 2
C H A P T E R 9Topics in Analytic Geometry
Section 9.1 Circles and Parabolas
772
! A parabola is the set of all points that are equidistant from a fixed line (directrix) and a fixed point(focus) not on the line.
! The standard equation of a parabola with vertex and
(a) Vertical axis and directrix is
(b) Horizontal axis and directrix is
! The tangent line to a parabola at a point makes equal angles with
(a) the line through and the focus.
(b) the axis of the parabola.
P
P
(y ! k)2 " 4p(x ! h), p # 0.
x " h ! p y " k
(x ! h)2 " 4p( y ! k), p # 0.
y " k ! p x " h
!h, k"
!x, y"
Vocabulary Check1. conic section 2. locus 3. circle, center
4. parabola, directrix, focus 5. vertex 6. axis
7. tangent
1.
x2 $ y2 " 18
x2 $ y2 " !#18"2 2.
x2 $ y2 " 32
x2 $ y2 " !4#2 "2
3.
!x ! 3"2 $ ! y ! 7"2 " 53
!x ! h"2 $ ! y ! k"2 " r2
" #4 $ 49 " #53
Radius " #!3 ! 1"2 $ !7 ! 0"2 4.
!x ! 6"2 $ ! y $ 3"2 " 113
!x ! h"2 $ ! y ! k"2 " r2
" #64 $ 49 " #113
Radius " #$6 ! !!2"%2 $ $!3 ! 4%2
5.
!x $ 3"2 $ ! y $ 1"2 " 7
!x ! h"2 $ ! y ! k"2 " r2
Diameter " 2#7 ! radius " #7 6.
!x ! 5"2 $ ! y $ 6"2 " 12
!x ! h"2 $ ! y ! k"2 " r2
Diameter " 4#3 ! radius " 2#3
7.
Center:
Radius: 7
!0, 0"
x2 $ y2 " 49 8.
Center:
Radius: 1
!0, 0"
x2 $ y2 " 1 9.
Center:
Radius: 4
!!2, 7"
!x $ 2"2 $ !y ! 7"2 " 16
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Page 3
Section 9.1 Circles and Parabolas 773
10.
Center:
Radius: 6
!!9, !1"
!x $ 9"2 $ !y $ 1"2 " 36 11.
Center:
Radius: #15
!1, 0"
!x ! 1"2 $ y2 " 15 12.
Center:
Radius: #24 " 2#6
!0, !12"
x2 $ !y $ 12"2 " 24
13.
Center:
Radius: 2
!0, 0"
x2 $ y2 " 4
14
x2 $14
y2 " 1 14.
Center:
Radius: 3
!0, 0"
x2 $ y2 " 9
19
x2 $19
y2 " 1 15.
Center:
Radius:#32
!0, 0"
x2 $ y2 "34
43
x2 $43
y2 " 1
16.
Center:
Radius:#23
!0, 0"
x2 $ y2 "29
92
x2 $92
y2 " 1 17.
Center:
Radius: 1
!1, !3"
!x ! 1"2 $ !y $ 3"2 " 1
!x2 ! 2x $ 1" $ !y2 $ 6y $ 9" " !9 $ 1 $ 9
18.
Center:
Radius: 3
!5, 3"
!x ! 5"2 $ !y ! 3"2 " 9
!x2 ! 10x $ 25" $ !y2 ! 6y $ 9" " !25 $ 25 $ 9 19.
Center:
Radius: 1
!!32, 3"
!x $ 32"2
$ ! y ! 3"2 " 1
4!x $ 32"2
$ 4!y ! 3"2 " 4
4!x2 $ 3x $ 94" $ 4!y2 ! 6y $ 9" " !41 $ 9 $ 36
20.
Center:
Radius: 103
!!3, 2"
!x $ 3"2 $ !y ! 2"2 " 1009
9!x $ 3"2 $ 9!y ! 2"2 " 100
9!x2 $ 6x $ 9" $ 9!y2 ! 4y $ 4" " !17 $ 81 $ 36
21.
Center:
Radius: 4
!0, 0"
x2 $ y2 " 16
!1!2!3!5 1 2 3 5
!2!3
!5
123
5
x
y x2 " 16 ! y2 22.
Center:
Radius: 9
!0, 0"
x2 $ y2 " 81
!2!4!6!10 2 4 6 8 10
!4!6!8
!10
2468
10
x
y y2 " 81 ! x2
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Page 4
774 Chapter 9 Topics in Analytic Geometry
23.
Center:
Radius: 3
!1!2!3!5!6!7 2 3
!2!3!4
!6!7
23
x
y!!2, !2"
!x $ 2"2 $ ! y $ 2"2 " 9
!x2 $ 4x $ 4" $ ! y2 $ 4y $ 4" " 1 $ 4 $ 4
x2 $ 4x $ y2 $ 4y ! 1 " 0 24.
Center:
Radius: 2!1!2 2 51 4 6 7 8
!2!3!4!5!6!7!8!9
1x
y!3, !3"
!x ! 3"2 $ ! y $ 3"2 " 4
!x2 ! 6x $ 9" $ ! y2 $ 6y $ 9" " !14 $ 9 $ 9
x2 ! 6x $ y2 $ 6y $ 14 " 0
25.
Center:
Radius: 5
!7, !4"
!x ! 7"2 $ ! y $ 4"2 " 25
!x2 ! 14x $ 4" $ ! y2 $ 8y $ 16" " !40 $ 49 $ 16
!2 4 6 8 10
!4!6!8
!10!12!14
246
x
y
14 16 18
x2 ! 14x $ y2 $ 8y $ 40 " 0
26.
Center:
Radius: 2
!1!2!3!4!5!6!7!8 1 2
123456789
x
y!!3, 6"
!x $ 3"2 $ ! y ! 6"2 " 4
!x2 $ 6x $ 9" $ ! y2 ! 12y $ 36" " !41 $ 9 $ 36
x2 $ 6x $ y2 ! 12y $ 41 " 0 27.
Center:
Radius: 6
!2!4!8!10 2 4 6 8 10
!4
!8!10
24
810
x
y!!1, 0"
!x $ 1"2 $ y2 " 36
!x2 $ 2x $ 1" $ y2 " 35 $ 1
x2 $ 2x $ y2 ! 35 " 0
28.
Center:
Radius: 4!1!2!3!4!5 1 2 3 4 5
!2!3!4!5!6!7!8
1x
y!0, !5"
x2 $ ! y2 $ 5"2 " 16
x2 $ ! y2 $ 10y $ 25" " !9 $ 25
x2 $ y2 $ 10y $ 9 " 0 29. intercepts:
intercepts:
!2, 0"
x " 2
!x ! 2"2 " 0
!x ! 2"2 $ !0 $ 3"2 " 9x-
!0, !3 ± #5 " y " !3 ± #5
! y $ 3"2 " 5
4 $ ! y $ 3"2 " 9
!0 ! 2"2 $ ! y $ 3"2 " 9y-
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Page 5
Section 9.1 Circles and Parabolas 775
30. intercepts:
intercepts:
!!2, 0", !!8, 0"
x " !8, !2
x $ 5 " ±3
!x $ 5"2 " 9
!x $ 5"2 $ 16 " 25
!x $ 5"2 $ !0 ! 4"2 " 25x-
!0, 4"
y " 4
! y ! 4"2 " 0
!0 $ 5"2 $ ! y ! 4"2 " 25y- 31. intercepts: Let
intercepts: Let
!1 ± 2#7, 0" x " 1 ± 2#7
x ! 1 " ±#28
!x ! 1"2 " 28
x2 ! 2x $ 1 " 27 $ 1
x2 ! 2x ! 27 " 0
y " 0.x-
!0, 9", !0, !3" y " 9, !3
y ! 3 " ±6
! y ! 3"2 " 36
y2 ! 6y $ 9 " 27 $ 9
y2 ! 6y ! 27 " 0
x " 0.y-
32. intercepts: Let
No solution
No intercepts
intercepts: Let
!!4 ± #7, 0" x " !4 ± #7
x $ 4 " ±#7
!x $ 4"2 " 7
x2 $ 8x $ 16 " !9 $ 16
x2 $ 8x $ 9 " 0
y " 0.x-
y-
y2 $ 2y $ 9 " 0
x " 0.y- 33. intercepts:
No solution
No intercepts
intercepts:
!6 ± #7, 0" x " 6 ± #7
x ! 6 " ±#7
!x ! 6"2 " 7
!x ! 6"2 $ !0 $ 3"2 " 16x-
y-
" !20
! y $ 3"2 " 16 ! 36
!0 ! 6"2 $ ! y $ 3"2 " 16y-
34. intercepts:
No solution
No intercepts
intercepts:
No solution
No interceptsx-
" !60
!x $ 7"2 " 4 ! 64
!x $ 7"2 $ !0 ! 8"2 " 4x-
y-
" !45
! y ! 8"2 " 4 ! 49
!0 $ 7"2 $ ! y ! 8"2 " 4y- 35. (a) Radius: 81; Center:
(b) The distance from to is
Yes, you would feel the earthquake.
(c)
You were miles from the outerboundary.
81 ! 75 " 6
x
y
!40 40
!40
40 (60, 45)
x2 + y2 = 812
#602 $ 452 " #5625 " 75 miles.
!0, 0"!60, 45"x2 $ y2 " 812 " 6561
!0, 0"
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Page 6
776 Chapter 9 Topics in Analytic Geometry
36. (a)
r & 23.937 feet
r "#1800%
r2 "1800
%
Area " %r2 " 1800 (b)
longer radius27.640 ! 23.937 & 3.703
R "#2400%
& 27.640 feet
%R2 " 2400
37.
Vertex:
Opens to the left since isnegative.
Matches graph (e).
p
!0, 0"
y2 " !4x 38.
Vertex:
Opens upward
Matches graph (b).
p " 12 > 0
!0, 0"
x2 " 2y 39.
Vertex:
Opens downward since isnegative.
Matches graph (d).
p
!0, 0"
x2 " !8y
40.
Vertex:
Opens to the left
Matches graph (f).
p " !3 < 0
!0, 0"
y2 " !12x 41.
Vertex:
Opens to the right since ispositive.
Matches graph (a).
p
!3, 1"
(y ! 1)2 " 4(x ! 3) 42.
Vertex:
Opens downward
Matches graph (c).
p " ! 12 < 0
!!3, 1"
!x $ 3"2 " !2!y ! 1"
43. Vertex:
Graph opens upward.
Point on graph:
! x2 " 32 y.
Thus, x2 " 4!38"y ! y " 2
3 x2
38 " p
9 " 24p
32 " 4p!6"
!3, 6"
x2 " 4py
!0, 0" ! h " 0, k " 0 44. Point:
y2 " !18x
x " ! 118 y2
! 118 " a
!2 " a!6"2
x " ay2
!!2, 6" 45. Vertex:
Focus:
x2 " !6y
x2 " 4!!32"y
!x ! h"2 " 4p!y ! k"
!0, !32" ! p " !3
2
!0, 0" ! h " 0, k " 0
46. Focus:
y2 " 10x
y2 " 4px " 4! 52"x
! 52, 0" ! p " 5
2 47. Vertex:
Focus:
y2 " !8x
y2 " 4!!2"x
!y ! k"2 " 4p!x ! h"
!!2, 0" ! p " !2
!0, 0" ! h " 0, k " 0 48. Focus:
x2 " 4y
x2 " 4py " 4!1"y
!0, 1" ! p " 1
49. Vertex:
Directrix:
x2 " 4y or y " 14x2
!x ! 0"2 " 4!1"!y ! 0"
!x ! h"2 " 4p!y ! k"
y " !1 ! p " 1
!0, 0" ! h " 0, k " 0 50. Directrix:
x2 " !12y
x2 " 4py
y " 3 ! p " !3 51. Vertex:
Directrix:
y2 " !8x
y2 " 4px
x " 2 ! p " !2
!0, 0" ! h " 0, k " 0
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Page 7
Section 9.1 Circles and Parabolas 777
52. Directrix:
y2 " 12x
y2 " 4px
x " !3 ! p " 3
53. Vertex:
Horizontal axis and passes through the point
y2 " 9x
y2 " 4!94"x
36 " 16p ! p " 94
62 " 4p!4"
y2 " 4px
!y ! 0"2 " 4p!x ! 0"
!y ! k"2 " 4p!x ! h"
!4, 6"
!0, 0" ! h " 0, k " 0 54. Vertical axis
Passes through
x2 " !3y
x2 " 4!!34"y
9 " !12p ! p " !34
!!3"2 " 4p!!3"
x2 " 4py
!!3, !3"
55.
Vertex:
Focus:
Directrix:
–1
1
2
3
4
5
–3 –2 2 3
y
x
y " !12
!0, 12"!0, 0"
x2 " 2y " 4! 12 "y; p " 1
2
y " 12 x2 56.
Vertex:
Focus:
Directrix:
!1!2 1 2
!2
!3
!4
x
y
y " 116
!0, ! 116"
!0, 0"
x2 " !14 y " 4!! 1
16"y, p " ! 116
y " !4x2 57.
Vertex:
Focus:
Directrix:
–6 –5 –4 –3 –2 –1 1 2
–4
–3
3
4
y
x
x " 32
!!32, 0"
!0, 0"
y2 " 4!!32"x; p " !3
2
y2 " !6x
58.
Vertex:
Focus:
Directrix:
!2 2 4 6
4
y
x
x " !34
!34, 0"!0, 0"
y2 " 4!34"x; p " 3
4
y2 " 3x 59.
Vertex:
Focus:
Directrix:
x
y
!4!6!8 4 6 8!2
!4
!6
!8
!10
4
6
2
y " 2
!0, !2"
!0, 0"
x2 " 4!!2"y; p " !2
x2 $ 8y " 0 60.
Vertex:
Focus:
Directrix:
–5 –4 –3 –2 –1 1
–3
–2
2
3
x
y
x " 14
!!14, 0"
!0, 0"
y2 " 4!!14"x, p " !1
4
y2 " !x
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Page 8
778 Chapter 9 Topics in Analytic Geometry
61.
Vertex:
Focus:
Directrix: y " !1
!!1, !5"x
y
!2
!4
!6
!8
!10
!12
4
2
2
!!1, !3"
h " !1, k " !3, p " !2
!x $ 1"2 " 4!!2"!y $ 3"
!x $ 1"2 $ 8!y $ 3" " 0 62.
Vertex:
Focus:
Directrix:
x
y
!1!2 1 2 3 4 5 6!1
!2
!3
!4
!5
!6
!7
1
x " 214
!5 ! 14, !4" " !19
4 , !4"!5, !4"
!y $ 4"2 " !!x ! 5" " 4!!14"!x ! 5"
!x ! 5" $ !y $ 4"2 " 0
63.
Vertex:
Focus:
Directrix: x " 0
!!4, !3"–10 –8 –6 –4
–8
–6
–4
–2
2
x
y!!2, !3"
!y $ 3"2 " 4!!2"!x $ 2"; p " !2
y2 $ 6y $ 8x $ 25 " 0 64.
Vertex:
Focus:
Directrix: x " !2
!0, 2"
–4 2 4
–2
4
6
x
y!!1, 2"
!y ! 2"2 " 4!x $ 1"; p " 1
y2 ! 4y ! 4x " 0
65.
Vertex:
Focus:
Directrix:
1 32!1!3 !2!4!5
!2
3
4
5
6
x
yy " 1
!!32, 2 $ 1" " !!3
2, 3"!!3
2, 2"h " !3
2, k " 2, p " 1!!x $ 32"2
" 4!y ! 2" 66.
Vertex:
Focus:
Directrix:
!1!2!3
!1
3
4
1 2x
yy " 0
!!12, 1 $ 1" " !!1
2, 2"!!1
2, 1"!x $ 1
2"2" 4!y ! 1" ! p " 1
67.
Vertex:
Focus:
Directrix: y " 0
!1, 2"
!1, 1"
h " 1, k " 1, p " 1
!x ! 1"2 " 4!1"!y ! 1"
–2 2 4
2
4
6
x
y 4y ! 4 " !x ! 1"2
y " 14!x2 ! 2x $ 5" 68.
Vertex:
Focus:
Directrix: x " 7
!9, !1"
!2
!4
!6
2
4
6
2 4 6 10 12x
y!8, !1"
!y $ 1"2 " 4!1"!x ! 8"
y2 $ 2y $ 1 " 4x ! 33 $ 1
4x ! y2 ! 2y ! 33 " 0
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Page 9
Section 9.1 Circles and Parabolas 779
69.
Vertex:
Focus:
Directrix:
2!1!3!4!6
!2
!3
3
2
1
4
5
x
yy " 52
!!2, 1 ! 32" " !!2, !1
2"!!2, 1"
!x $ 2"2 " 4!!32"!y ! 1"
!x $ 2"2 " !6!y ! 1"
x2 $ 4x $ 4 " !6y $ 2 $ 4 " !6y $ 6
x2 $ 4x $ 6y ! 2 " 0 70.
Vertex:
Focus:
Directrix: y " 1
!1,!3"!1!2
!2
2
!3!4!5!6!7!8
2 3 4 5 6!3!4x
y!1, !1"
!x ! 1"2 " !8!y $ 1" " 4!!2"!y $ 1"
x2 ! 2x $ 1 " !8y ! 9 $ 1
x2 ! 2x $ 8y $ 9 " 0
71.
Vertex:
Focus:
Directrix:
To use a graphing calculator, enter:
y2 " !12 ! #1
4 ! x
y1 " !12 $ #1
4 ! x
x " 12
!0, !12"
!1 1!2
!2
1
2
!3x
y!14, !1
2"h " 1
4, k " !12, p " !1
4
!y $ 12"2 " 4!!1
4"!x ! 14"
y2 $ y $ 14 " !x $ 1
4
y2 $ x $ y " 0 72.
Vertex:
Focus:
Directrix: x " !2
!0, 0"
!4
!4
!2
2
4
6
!6
2 4 6 8x
y!!1, 0"
y2 " 4x $ 4 " 4!1"!x $ 1"
y2 ! 4x ! 4 " 0
73. Vertex:opens downward
Passes through:
!x ! 3"2 " !!y ! 1"
" !!x ! 3"2 $ 1
" !x2 $ 6x ! 8
y " !!x ! 2"!x ! 4"
!2, 0", !4, 0"
!3, 1", 74. Vertex:
Passes through:
!y ! 3"2 " !2!x ! 5"
p " ! 12
1 " 4p!4.5 ! 5"
!y ! 3"2 " 4p!x ! 5"
!y ! k"2 " 4p!x ! h"
!4.5, 4"
k " 3!5, 3" ! h " 5, 75. Vertex:opens to the right
Focus:
y2 " 2!x $ 2"
y2 " 4!12"!x $ 2"
12 " p
!!32, 0"
!!2, 0",
76. Vertex:
Focus:
!x ! 3"2 " 3!y $ 3"
!x ! h"2 " 4p!y ! k"
!3, !94" ! p " 3
4
k " !3!3, !3" ! h " 3, 77. Vertex:
Focus:
Horizontal axis:
!y ! 2"2 " !8!x ! 5"
!y ! 2"2 " 4!!2"!x ! 5"
p " 3 ! 5 " !2
!3, 2"
!5, 2"
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Page 10
780 Chapter 9 Topics in Analytic Geometry
81. Focus:
Directrix:
Horizontal axis
Vertex:
!y ! 2"2 " 8x
!y ! 2"2 " 4!2"!x ! 0"
p " 2 ! 0 " 2
!0, 2"
x " !2
!2, 2" 82. Focus:
Directrix:
Vertex:
x2 " !8!y ! 2"
x2 " 4!!2"!y ! 2"
!0, 2"
y " 4 ! p " !2
!0, 0"
83.
The point of tangency is!2, 4".
y2 " !#8x
!3
!6 6
5 y1 " #8x
y2 " 8x and x ! $ 2y3 " x $ 2
y2 ! 8x " 0x and 3x ! y $ 2 " 0 84.
The point of tangency is!6, !3".
!4
!4 8
4 y1 " ! 112 x2
12y " !x2 y2 " 3 ! x
x2 $ 12y " 0 and x $ y ! 3 " 0
85. focus:
Following Example 4, we find the intercept
Tangent line
Let intercept !2, 0".y " 0 ! x " 2 ! x-
y " 4x ! 8,
m "8 ! !!8"
4 ! 0" 4
b " !8!12
! b "172
!d1 " d2
d2 "#!4 ! 0"2 $ '8 !12(
2"
172
d1 "12
! b
!0, b".y-
'0, 12(!4, 8", p "
12
,x2 " 2y,
78. Vertex:
Focus:
!x $ 1"2 " !8!y ! 2"
!x $ 1"2 " 4!!2"!y ! 2"
!x ! h"2 " 4p!y ! k"
!!1, 0" ! p " !2
k " 2!!1, 2" ! h " !1, 79. Vertex:
Directrix:
Vertical axis
x2 " 8!y ! 4"
!x ! 0"2 " 4!2"!y ! 4"
p " 4 ! 2 " 2
y " 2
!0, 4" 80. Vertex:
Directrix:
!y ! 1"2 " !12!x $ 2"
!y ! 1"2 " 4!!3"!x ! !!2""
!y ! k"2 " 4p!x ! h"
x " 1 ! p " !3
k " 1!!2, 1" ! h " !2,
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Page 11
Section 9.1 Circles and Parabolas 781
86.
Focus:
Tangent line:
-intercept: '!32
, 0(x
y " !3x !92
! 6x $ 2y $ 9 " 0
m "!!9)2" ! !9)2"
0 $ 3" !3
b " !92
12
! b " 5
d2 "#!!3 ! 0"2 $ '92
!12(
2
" 5
d1 "12
! b
'0, 12(
p "12
4'12(y " x2
2y " x2 87.
Focus:
Following Example 4, we find the intercept
Let intercept '!12
, 0(.y " 0 ! x " !12
! x-
y " 4x $ 2
m "!2 ! 2!1 ! 0
" 4
b " 2!18
$ b "178
!d1 " d2
d2 "#!!1 ! 0"2 $ '!2 $18(
2"
178
d1 "18
$ b
!0, b".y-
'0, !18(
! p " !18
y " !2x2 ! x2 " !12
y " 4'!18(y
88.
Focus:
Intercept: !1, 0"
y " !8x $ 8
m "!8 ! 82 ! 0
" !8
d1 " d2 ! 18
$ b "658
! b " 8
d2 "#!2 ! 0"2 $ '!8 $18(
2"
658
d1 "18
$ b
'0, !18(
x2 " !12
y " 4'!18(y ! p " !
18
y " !2x2, !2, !8" 89.
is a maximum of $23,437.50 when televisions.
00 250
25,000
x " 125R
R " 375x !32
x2
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Page 12
782 Chapter 9 Topics in Analytic Geometry
90. (a)
y "x2
12,288
x2 " 4!3072"y
3072 " p
1024 "13
p
322 " 4p' 112(
x2 " 4py 91. (a)
(b) When
Depth: inches83
y "166
"83
.
6y " 16
x " 4,
!or y2 " 6x"
x2 " 4'32(y " 6y
x
y
!1!2!3!4 1 2 3 4!1
!2
1
2
4
5
6
320,( (
8 in.
x2 " 4py, p "32
(b)
x & 22.6 feet
512 " x2
12,288
24" x2
124
"x2
12,288
92. on parabola
The wire should be insertedinches from the bottom.9
4
p " 3616 " 9
4
36 " 4p!4"
x
y
!2!4!6 2 4 6!2
2
6
8
10
(6, 4)
x2 " 4py, !6, 4" 93. (a)
(c)
y
x
(!640, 152) (640, 152)
(b)
y " 1951,200 x2
p " 12,80019
6402 " 4p!152"
x2 " 4py
x 0 200 400 500 600
y 0 14.84 59.38 92.77 133.59
94. (a) passes through point
or
(b) !0.1 " ! 1640 x2 ! x " 8 feet
y " ! 1640 x2 x2 " !640y
x2 " 4!!160"y
256 " 4p!!25" ! p " !160
!16, !25".x2 " 4py 95. Vertex:
Point:
y2 " 640x
y2 " 4!160"x
8002 " 4p!1000% ! p " 160
!1000, 800"
y2 " 4px
!0, 0%
96. (a)
(b)
! ! 0"x2 " !16,400!y ! 4100"
!x ! 0"2 " 4!!4100"!y ! 4100"
p " !4100, !h, k" " !0, 4100"
V " 17,500#2 mi)hr & 24,750 mi)hr 97.
(a)
(b) The highest point is at Thedistance is the -intercept of feet.&15.69x
!6.25, 7.125".
00 16
10
y " !0.08x2 $ x $ 4
!12.5y $ 89.0625 " x2 ! 12.5x $ 39.0625
!12.5! y ! 7.125" " !x ! 6.25"2
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Page 13
Section 9.1 Circles and Parabolas 783
98. (a)
(b)
! x & 69.3 ft
y " 0 " !164
x2 $ 75 ! x2 " 75!64"
" !164
x2 $ 75
" !16x2
322 $ 75
y " !16x2
v2 $ s
x2 " !116
v2!y ! s" 99. The slope of the line joining and the centeris The slope of the tangent line at is Thus,
3x ! 4y " 25, tangent line.
4y $ 16 " 3x ! 9
y $ 4 "34
!x ! 3"
34.!3, !4"!4
3.!3, !4"
100. The slope of the line joining and the center is The slope of the tangent line at is Thus,
5x ! 12y $ 169 " 0, tangent line.
12y ! 144 " 5x $ 25
y ! 12 "512
!x $ 5"
512.!!5, 12"
!125 .
!!5, 12" 101. The slope of the line joining and the center is The slope ofthe tangent line is Thus,
#2x ! 2y " 6#2, tangent line.
2y $ 4#2 " #2x ! 2#2
y $ 2#2 "#22
!x ! 2"
1)#2 " #2)2.!!2#2 ")2 " !#2.
!2, !2#2 "
102. The slope of the line joining and the center is The slope of the tangent line is Thus,
#5x ! y $ 12 " 0, tangent line.
y ! 2 " #5x $ 10
y ! 2 " #5!x $ 2#5 "#5.
2)!!2#5 " " !1)#5.!!2#5, 2"
103. False. The center is !0, !5". 104. True 105. False. A circle is a conic section.
106. False. A parabola cannotintersect its directrix or focus.
107. True 108. False. The directrix is below the axis.x-
y " !14
109. Answers will vary. See the reflective property of parabolas, page 599.
110. The graph of is a single point,
The plane intersects the double-napped cone at the vertices of the cones.
!1!2!3!4!5 1 2 3 4 5
!2!3!4!5
12345
x
y
!0, 0".x2 $ y2 " 0
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Page 14
784 Chapter 9 Topics in Analytic Geometry
111.
For the upper half of the parabola,
y ! !6"x " 1# " 3.
y # 3 ! !6"x " 1#
"y # 3#2 ! 6"x " 1# 112.
For the lower half of the parabola,
y ! #1 # !2"x # 2#.
y " 1 ! #!2"x # 2#
"y " 1#2 ! 2"x # 2#
113.
Relative maximum:
Relative minimum: "0.67, 0.22#
"#0.67, 3.78#
f "x# ! 3x3 # 4x " 2 114.
Relative minimum: at x ! #0.75#1.13
f "x# ! 2x2 " 3x
115.
Relative minimum: "#0.79, 0.81#
f "x# ! x4 " 2x " 2 116.
Relative minimum: at 0.88
Relative maximum: 1.11 at #0.88
#3.11
f "x# ! x5 # 3x # 1
Section 9.2 Ellipses
! An ellipse is the set of all points the sum of whose distances from two distinct fixed points (foci)is constant.
! The standard equation of an ellipse with center and major and minor axes of lengths and is
(a) if the major axis is horizontal.
(b) if the major axis is vertical.
! where is the distance from the center to a focus.
! The eccentricity of an ellipse is e !ca
.
cc2 ! a2 # b2
"x # h#2
b2 ""y # k#2
a2 ! 1
"x # h#2
a2 ""y # k#2
b2 ! 1
2b2a"h, k#
"x, y#
Vocabulary Check
1. ellipse 2. major axis, center
3. minor axis 4. eccentricity
1.
Center:
Vertical major axis
Matches graph (b).
a ! 3, b ! 2
"0, 0#
x2
4"
y2
9! 1 2.
Center:
Horizontal major axis
Matches graph (c).
a ! 3, b ! 2
"0, 0#
x2
9"
y2
4! 1 3.
Center:
Vertical major axis
Matches graph (d).
a ! 5, b ! 2
"0, 0#
x2
4"
y2
25! 1
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Page 15
Section 9.2 Ellipses 785
4.
Center:
Horizontal major axis
Matches graph (f).
a ! 2, b ! 1
"0, 0#
x2
4" y2 ! 1 5.
Center:
Horizontal major axis
Matches graph (a).
a ! 4, b ! 1
"2, #1#
"x # 2#2
16" "y " 1#2 ! 1 6.
Center:
Horizontal major axis
Matches graph (e).
"#2, #2#
"x " 2#2
9"
"y " 2#2
4! 1
7.
Center:
Vertices:
Foci:
e !ca
!!55
8
"±!55, 0#"±8, 0#
c ! !64 # 9 ! !55
a ! 8, b ! 3,
"0, 0#
!2!4!10 2 4 10
!4!6!8
!10
2468
10
x
yx2
64"
y2
9! 1 8.
Center:
Vertices:
Foci:
e !ca
!!65
9
"0, ±!65#"0, ±9#
c ! !81 # 16 ! !65
a ! 9, b ! 4,
"0, 0#
!2!6!8!10 2 6 8 10
!4!6
!10
246
10
x
yx2
16"
y2
81! 1
9.
Center:
Vertices:
Foci:
x
y
!2!4 2 6 10!2
!4
!6
!8
2
4
6
e !ca
!35
"4, #1 ± 3#; "4, #4#, "4, 2#
"4, #1 ± 5#; "4, #6#, "4, 4#
a ! 5, b ! 4, c ! 3
"4, #1#
"x # 4#2
16"
"y " 1#2
25! 1 10.
Center:
Foci:
Vertices:
!1!3 !2!4!7 1
!2
1
2
3
4
6
x
ye !ca
!24
!12
"#3, 2 ± 4#; "#3, #2#, "#3, 6#
"#3, 2 ± 2#; "#3, 0#, "#3, 4#
a ! 4, b ! 2!3, c ! !16 # 12 ! 2
"#3, 2#
"x " 3#2
12"
"y # 2#2
16! 1
11.
Center:
Foci:
Vertices:
e !!5$23$2
!!53
%#5 #32
, 1& ! %#132
, 1&%#5 "32
, 1& ! %#72
, 1&,
%#5 "!52
, 1&, %#5 #!52
, 1&
a !32
, b ! 1, c !!94
# 1 !!52
"#5, 1#
1!1!3 !2!4!5!6!7
!2
!3
!4
4
2
3
1
x
y"x " 5#2
9$4" "y # 1#2 ! 1
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Page 16
786 Chapter 9 Topics in Analytic Geometry
12.
Center:
Foci:
Vertices:
Eccentricity:!32
"#3, #4#, "#1, #4#
%#2 "!32
, #4&, %#2 #!32
, #4&"#2, #4#
a ! 1, b !12
, c ! !a2 # b2 !!32
–3 –2 –1 1
–5
–4
–3
–2
–1
x
y"x " 2#2 ""y " 4#2
1$4! 1
13. (a)
(b)
Center:
Vertices:
Foci:
e !ca
!4!2
6!
2!23
"±4!2, 0#"±6, 0#
"0, 0#
a ! 6, b ! 2, c ! !36 # 4 ! !32 ! 4!2
x2
36"
y2
4! 1
x2 " 9y2 ! 36 (c)
!8!10 6 8 10
!4!6!8
!10
468
10
x
y
14. (a)
(b)
Center:
Vertices:
Foci:
e !ca
!!15
4
"0, ±!15#"0, ±4#
"0, 0#
a ! 4, b ! 1, c ! !16 # 1 ! !15
x2 "y2
16! 1
16x2 " y2 ! 16 (c)
!2!3!4!5 2 3 4 5
!4!5
1
45
x
y
15. (a) (c)
(b)
Center:
Foci:
Vertices:
e !!53
"#2, 6#, "#2, 0#
"#2, 3 ± !5 #"#2, 3#
a ! 3, b ! 2, c ! !5
"x " 2#2
4"
"y # 3#2
9! 1
9"x2 " 4x " 4# " 4"y2 # 6y " 9# ! #36 " 36 " 36
1 2!1!3 !2!4!5!6
!2
4
6
2
3
x
y 9x2 " 4y2 " 36x # 24y " 36 ! 0
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Page 17
Section 9.2 Ellipses 787
16. (a) (c)
(b)
Center:
Foci:
Vertices:
e !2!5
6!
!53
"3, #5 ± 6#; "3, 1#, "3, #11#
"3, #5 ± 2!5#"3, #5#
a ! 6, b ! 4, c ! !20 ! 2!5
"x # 3#2
16"
"y " 5#2
36! 1
9"x # 3#2 " 4"y " 5#2 ! 144!2 2 4 10 128
2
!10
!12
y
x
9"x2 # 6x " 9# " 4"y2 " 10y " 25# ! #37 " 81 " 100
17. (a) (c)
(b)
Center:
Foci:
Vertices:
e !!2!3
!!63
%#32
, 52
± 2!3&%#
32
, 52
± 2!2&%#
32
, 52&
a ! 2!3, b ! 2, c ! 2!2
"x " 3
2#2
4"
"y # 52#2
12! 1
6%x "32&
2" 2%y #
52&
2! 24
6%x2 " 3x "94& " 2%y2 # 5y "
254 & ! #2 "
272
"252
2
2
4
!4
!2
!6x
y 6x2 " 2y2 " 18x # 10y " 2 ! 0
18. (a) (c)
(b)
Center:
Foci:
Vertices:
e !!32
%9, #52&, %#3, #
52&
%3 ± 3!3, #52&
%3, #52&
a ! 6, b ! 3, c ! !36 # 9 ! !27 ! 3!3
"x # 3#2
36"
"y " 52#2
9! 1
"x # 3#2 " 4%y "52&
2! 36
!3 !1!2!3!4
!6!7!8
21 3 4 5 6 9 10
21
3456
x
y "x2 # 6x " 9# " 4%y2 " 5y "254 & ! 2 " 9 " 25
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Page 18
788 Chapter 9 Topics in Analytic Geometry
19. (a) (c)
(b)
Center:
Foci:
Vertices:
e !35
%94
, #1&, %#14
, #1&%7
4, #1&, %1
4, #1&
"1, #1#
a !54
, b ! 1, c !34
(x # 1)2
25$16" (y " 1)2 ! 1
16"x2 # 2x " 1# " 25"y2 " 2y " 1# ! #16 " 16 " 25
–2 –1 1 3
–3
–2
1
2
x
y 16x2 " 25y2 # 32x " 50y " 16 ! 0
20. (a) (c)
(b) Degenerate ellipse with center as the only point"2, 1#
9"x # 2#2 " 25"y # 1#2 ! 0
9"x2 # 4x " 4# " 25"y2 # 2y " 1# ! #61 " 36 " 25
1 2
1
2
x
y 9x2 " 25y2 # 36x # 50y " 61 ! 0
21. (a) (c)
(b)
Center:
Vertices:
Foci:
Eccentricity:ca
!!2!5
!!10
5
%12
± !2, #1&%1
2± !5, #1&
%12
, #1&a ! !5, b ! !3, c ! !5 # 3 ! !2
"x # 1
2#2
5"
"y " 1#2
3! 1
12%x #12&
2
" 20"y " 1#2 ! 60
12%x2 # 1 "14& " 20"y2 " 2y " 1# ! 37 " 3 " 20
x
y
!1!2!3 1 2 3
1
2
!2
!3
!4
12x2 " 20y2 # 12x " 40y # 37 ! 0
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Page 19
Section 9.2 Ellipses 789
23. Center:
Vertical major axis
x2
4"
y2
16! 1
a ! 4, b ! 2
"0, 0# 24. Vertices:
Endpoints of minor axis:
x2
4"
4y2
9! 1
x2
22 "y2
"3$2#2 ! 1
x2
a2 "y2
b2 ! 1
%0, ±32& ! b !
32
"±2, 0# ! a ! 2
25. Center:
Horizontal major axis
x2
9"
y2
5! 1
c ! 2 ! b ! !9 # 4 ! !5a ! 3,
"0, 0# 26. Vertices:
Foci:
Center:
y2
64"
x2
48! 1
"y # k#2
a2 ""x # h#2
b2 ! 1
"0, 0# ! "h, k#
b2 ! a2 # c2 ! 64 # 16 ! 48
"0, ±4# ! c ! 4
"0, ±8# ! a ! 8
27. Center:
Horizontal major axis
x2
16"
y2
7! 1
a ! 4 ! b ! !16 # 9 ! !7
c ! 3
"0, 0# 28. Center:
Horizontal major axis
x2
36"
y2
32! 1
a ! 6 ! b ! !36 # 4 ! !32 ! 4!2
c ! 2
"0, 0#
22. (a)
(c)
x
y
!1!2 1 2
!1
1
3
"x " 2
3#2
14
""y # 2#2
1! 1
36%x "23&
2
" 9"y # 2#2 ! 9
36%x2 "43
x "49& " 9"y2 # 4y " 4# ! #43 " 16 " 36
36x2 " 9y2 " 48x # 36y " 43 ! 0 (b)
Center:
Vertices:
Foci:
Eccentricity:ca
!!32
%#23
, 2 ±!32 &
%#23
, 2 ± 1& ! %#23
, 1&, %#23
, 3&%#
23
, 2&
c !!1 #14
!!32
b !12
,a ! 1,
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Page 20
790 Chapter 9 Topics in Analytic Geometry
29. Vertices:
Center:
Vertical major axis
Point:
21x2
400"
y2
25! 1
x2
400$21"
y2
25! 1
40021
! b2
400 ! 21b2
16b2 ! 1 #
425
!2125
42
b2 "22
25! 1
"4, 2#
x2
b2 "y2
25! 1
"x # h#2
b2 ""y # k#2
a2 ! 1
"0, 0#
"0, ±5# ! a ! 5 30. Vertical major axis
Passes through: and
x2
4"
y2
16! 1
x2
b2 "y2
a2 ! 1
a ! 4, b ! 2
"2, 0#"0, 4#
31. Center:
Vertical major axis
"x # 2#2
1"
"y # 3#2
9! 1
"x # h#2
b2 ""y # k#2
a2 ! 1
a ! 3, b ! 1
"2, 3# 32. Vertices:
Center:
Endpoints of minor axis:
"x # 2#2
4"
"y " 1#2
1! 1
"x # h#2
a2 ""y # k#2
b2 ! 1
"2, 0#, "2, #2# ! b ! 1
"2, #1# ! h ! 2, k ! #1
"0, #1#, "4, #1# ! a ! 2
33. Center:
Horizontal major axis
"x # 4#2
16"
" y # 2#2
1! 1
a ! 4, b ! 1 ! c ! !16 # 1 ! !15
"4, 2# 34. Center:
Horizontal major axis
"x # 2#2
9"
y2
5! 1
c ! 2, a ! 3 ! b2 ! a2 # c2 ! 9 # 4 ! 5
"2, 0#
35. Center:
Vertical major axis
x2
308"
" y # 4#2
324! 1
c ! 4, a ! 18 ! b2 ! a2 # c2 ! 324 # 16 ! 308
"0, 4#
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Page 21
Section 9.2 Ellipses 791
36. Center:
Vertex:
Minor axis length:
"x # 2#2 "4"y " 1#2
9! 1
"x # 2#2
1"
"y " 1#2
"3$2#2 ! 1
"x # h#2
b2 ""y # k#2
a2 ! 1
2 ! b ! 1
%2, 12& ! a !
32
"2, #1# ! h ! 2, k ! #1 37. Vertices:
Center:
Minor axis of length
Vertical major axis
"x # 3#2
9"
"y # 5#2
16! 1
"x # h#2
b2 ""y # k#2
a2 ! 1
6 ! b ! 3
"3, 5#
"3, 1#, "3, 9# ! a ! 4
38. Center:
Foci:
"x # 3#2
36"
"y # 2#2
32! 1
"x # h#2
a2 ""y # k#2
b2 ! 1
b2 ! a2 # c2 ! 36 # 4 ! 32
"1, 2#, "5, 2# ! c ! 2, a ! 6
a ! 3c
"3, 2# ! "h, k# 39. Center:
Vertices:
Horizontal major axis
x2
16"
"y # 4#2
12! 1
"x # h#2
a2 ""y # k#2
b2 ! 1
22 ! 42 # b2 ! b2 ! 12
a ! 2c ! 4 ! 2c ! c ! 2
"#4, 4#, "4, 4# ! a ! 4
"0, 4#
43.
e !ca
!2!2
3
a ! 3, b ! 1, c ! !9 # 1 ! 2!2
"x # 5#2
9"
" y " 2#2
1! 1
"x # 5#2 " 9" y " 2#2 ! 9
"x2 # 10x " 25# " 9" y2 " 4y " 4# ! #52 " 25 " 36
x2 " 9y2 # 10x " 36y " 52 ! 0
40. Vertices:
Endpoints of minor axis:
Center:
"x # 5#2
25"
"y # 6#2
36! 1
"x # h#2
b2 ""y # k#2
a2 ! 1
"5, 6# ! h ! 5, k ! 6
"0, 6#, "10, 6# ! b ! 5
"5, 0#, "5, 12# ! a ! 641.
e !ca
!!53
c ! !9 # 4 ! !5
a ! 3, b ! 2,
x2
4"
y2
9! 1 42.
e !ca
!!11
6
c ! !36 # 25 ! !11
a ! 6, b ! 5,
x2
25"
y2
36! 1
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Page 22
792 Chapter 9 Topics in Analytic Geometry
44.
e !ca
!12
a ! 2, b ! !3, c ! !4 # 3 ! 1
"x # 1#2
3"
" y " 3#2
4! 1
4"x # 1#2 " 3" y " 3#2 ! 12
4"x2 # 2x " 1# " 3" y2 " 6y " 9# ! #19 " 4 " 27
4x2 " 3y2 # 8x " 18y " 19 ! 0
45. Vertices:
Eccentricity:
Center:
Horizontal major axis
x2
25"
y2
9! 1
"0, 0#
b2 ! a2 # c2 ! 25 # 16 ! 9
45
!ca
! c !45
a ! 4
"±5, 0# ! a ! 5 46. Vertices:
Eccentricity:
x2
48"
y2
64! 1
x2
b2 "y2
a2 ! 1
b2 ! a2 # c2 ! 64 # 16 ! 48
c ! 4
12
!c8
e !12
!ca
h ! 0, k ! 0"0, ±8# ! a ! 8,
47. (a)
!20!40 20 40
!20
20
60
80
(!50, 0) (50, 0)
(0, 40)
x
y (b) Vertices:
Height at center:
Horizontal major axis
x2
2500"
y2
1600! 1, y " 0
x2
a2 "y2
b2 ! 1
40 ! b ! 40
"±50, 0# ! a ! 50 (c) For
The height five feet from the edge of the tunnel is approximately 17.44 feet.
y ' 17.44
y2 ! 304
y2 ! 1600%1 #452
2500&
452
2500"
y2
1600! 1.x ! 45,
48. (a)
!4!8!12!20 4 8 12 16 20
!8!12!16!20
48
1620
x
y
(0, 12)
(!16, 0) (16, 0)
(b)
x2
256"
y2
144! 1, y " 0
a ! 16, b ! 12 (c) When
Hence, the truck will be ableto drive through without crossing the center line.
y ' 9.4 > 9.
y2 ! 144!1 #102
256
x ! 10,
49. Let be the equation of the ellipse. Then and
Thus, the tacks are placed
at The string has a length of 2a ! 6 feet."±!5, 0#.c2 ! a2 # b2 ! 9 # 4 ! 5.a ! 3 !
b ! 2x2
a2 "y2
b2 ! 1
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Page 23
Section 9.2 Ellipses 793
50.
Distance between foci: feet2"4.7# ' 85.4
a !972
, b ! 23, c !!%972 &2
# "23#2 ' 4.7
%or x2
232 "y2
"97$2#2 ! 1&x2
"97$2#2 "y2
232 ! 1
x
y
!20 20 40
!40
40
51.
Length of major axis: 2a ! 2"20# ! 40 units
a ! 20
$a"10# ! 200
$a"10# ! 2$"10#2
$ab ! 2$r2
Area of ellipse ! 2"area of circle#
52. Center:
Ellipse:x2
321.84"
y2
19.02! 1
c2 ! a2 # b2 ! b2 ! a2 # c2 ' 19.02
e !ca
! 0.97 !c
17.94 ! c ! 17.4018
2a ! 35.88 ! a ! 17.94 ! a2 ' 321.84
"0, 0#, e ! 0.97 53.
x2
4.8841"
y2
1.3872! 1
b2 ! a2 # c2 ! b2 ! 1.3872
2a ! 4.42 ! a ! 2.21 ! c ! 1.87
a # c ! 0.34
a " c ! 4.08
54.
e !ca
' 0.0516
! 359.5
c ! 7325 # 6965.5
a ! 6965.5
x
b
ac!a
!b
y
a ! c
a + c
2a ! 13,931
a # c ! 228 " 6378 ! 6606
a " c ! 947 " 6378 ! 7325 55. For we have
When
! 2y !2b2
a.
! y2 !b4
a2
c2
a2 "y2
b2 ! 1 ! y2 ! b2%1 #a2 # b2
a2 &x ! c,
c2 ! a2 # b2.x2
a2 "y2
b2 ! 1,
56.
Points on the ellipse:
Length of latus recta:
Additional points: %!3, ±12&, %#!3, ±
12&
2b2
a! 1
"±2, 0#, "0, ±1#
a ! 2, b ! 1, c ! !3
!1
!2
2
1x
(
(
(
(
, !
,
!1
1
1
1
2
2
2
2
)
)
)
)
! 3
! 3
3,
3,
yx2
4"
y2
1! 1 57.
Points on the ellipse:
Length of latus recta:
Additional points: %±94
, #!7&, %±94
, !7&
2b2
a!
2"3#2
4!
92
"±3, 0#, "0, ±4#
a ! 4, b ! 3, c ! !7
x
!
!
9
9 9
94
4 4
4,
, ,
7
7 7! !
, 7(
( (
()
) )
)
y
!2!4 2 4
!2
2
x2
9"
y2
16! 1
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Page 24
794 Chapter 9 Topics in Analytic Geometry
58.
Points on the ellipse:
Length of latus recta:
Additional points: %±43
, #!5&, %±43
, !5&
2b2
a!
2 % 22
3!
83
"±2, 0#, "0, ±3#
x2
4"
y2
9! 1
!1!3
!2
2
1 3x
( (
((
, ,! 5 ! 5
5, 5 ,
4
4
4
4
3
3
3
3
) )
))
!
!
y9x2 " 4y2 ! 36 59.
Points on the ellipse:
Length of latus recta:
Additional points: %±3!5
5, #!2&, %±
3!55
, !2&
2b2
a!
2 % 3!5
!6!5
5
"±!3, 0#, "0, ±!5#
c ! !2
a ! !5, b ! !3,
x2
3"
y2
5! 1
!4 !2 2 4
!4
4
x
(
((
( , 2
, 2!, 2!
, 2 3 5
3 53 5
3 55
55
5 )
))
)
!
!
y 5x2 " 3y2 ! 15
60. Answers will vary. 61. True. If then the ellipse is elongated, notcircular.
e ' 1
62. True. The ellipse is inside the circle. 63. (a) The length of the string is
(b) The path is an ellipse because the sum of thedistances from the two thumbtacks is alwaysthe length of the string, that is, it is constant.
2a.
64. (a)
(b)
by the Quadratic Formula
Since we choose
x2
196"
y2
36! 1
x2
142 "y2
62 ! 1
a ! 14 and b ! 6.a > b,
b ! 64 ORb ! 14
a ' 14 or a ' 6
$a2 # 20$a " 264 ! 0
264 ! $a"20 # a#
A ! $ab ! $a"20 # a#
a " b ! 20 ! b ! 20 # a (c)
(d)
The area is maximum when and it is a circle.
a ! b ! 10
00 24
360
8 9 10 11 12 13
301.6 311.0 314.2 311.0 301.6 285.9A
a
65. Center:
Foci:
Horizontal major axis
"x # 6#2
324"
" y # 2#2
308! 1
b2 ! a2 # b2 ! b ! !182 # 16 ! !308
"a " c# " "a # c# ! 2a ! 36 ! a ! 18
"2, 2#, "10, 2# ! c ! 4
"6, 2# 66.
The sum of the distancesfrom any point on theellipse to the two foci isconstant. Using the vertex
you have
From the figure,
2!b2 " c2 ! 2a ! a2 ! b2 " c2.
"a " c# " "a # c# ! 2a.
"a, 0#,
x
b
b
acc
!c!a
!b
y
b2 + c2
x2
a2 "y2
b2 ! 1
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Page 25
Section 9.3 Hyperbolas 795
67. Arithmetic: d ! "11 68. Geometric: r ! 12 69. Geometric: r ! 2 70. Arithmetic: d ! 1
71. !6
n!0 3n ! 1093 72. !
6
n!0 ""3#n ! 547 73. !
10
n!1 4"3
4#n"1$ 15.099 74. !10
n!0 5"4
3#n $ 340.155
Section 9.3 Hyperbolas
! A hyperbola is the set of all points the difference of whose distances from two distinct fixed points(foci) is constant.
! The standard equation of a hyperbola with center and transverse and conjugate axes of lengths and is:
(a) if the transverse axis is horizontal.
(b) if the transverse axis is vertical.
! where is the distance from the center to a focus.
! The asymptotes of a hyperbola are:
(a) if the transverse axis is horizontal.
(b) the transverse axis is vertical.
! The eccentricity of a hyperbola is
! To classify a nondegenerate conic from its general equation (a) If then it is a circle.(b) If but not both), then it is a parabola.(c) If then it is an ellipse.(d) If then it is a hyperbola.AC < 0,
AC > 0, AC ! 0 (A ! 0 or C ! 0, A ! C (A # 0, C # 0),
Ax2 $ Cy2 $ Dx $ Ey $ F ! 0:
e !ca
.
y ! k ±ab
"x " h#
y ! k ±ba
"x " h#
cc2 ! a2 $ b2
"y " k#2
a2 ""x " h#2
b2 ! 1
"x " h#2
a2 ""y " k#2
b2 ! 1
2b2a"h, k#
"x, y#
Vocabulary Check1. hyperbola 2. branches 3. transverse axis, center
4. asymptotes 5. Ax2 $ Cy2 $ Dx $ Ey $ F ! 0
1. Center:
Vertical transverse axis
Matches graph (b).
a ! 3, b ! 5, c ! %34
"0, 0# 2. Center:
Vertical transverse axis
Matches graph (c).
a ! 5, b ! 3
"0, 0#
3. Center:
Horizontal transverse axis
Matches graph (a).
a ! 4, b ! 2
"1, 0# 4. Center:
Horizontal transverse axis
Matches graph (d).
a ! 4, b ! 3
""1, 2#©H
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Page 26
796 Chapter 9 Topics in Analytic Geometry
5.
Center:
Vertices:
Foci:
Asymptotes: y ! ±x
"±%2, 0#"±1, 0#
"0, 0#
a ! 1, b ! 1, c ! %2
–2 2
–2
–1
1
2
x
yx2 " y2 ! 1 6.
Center:
Vertices:
Foci:
Asymptotes: y ! ±ba
x ! ±53
x
"±%34, 0#"±3, 0#
c ! %32 $ 52 ! %34
a ! 3, b ! 5,
"0, 0#
!4
!6!8
!10
!6!8 2 4 6 8 10
468
10
x
yx2
9"
y2
25! 1
7.
Center:
Vertices:
Foci:
Asymptotes: y ! ±12
x
"0, ±%5 #"0, ±1#
"0, 0#
a ! 1, b ! 2, c ! %5
–3 –2 2 3
–3
–2
2
3
y
x
y2
1"
x2
4! 1 8.
Center:
Vertices:
Foci:
Asymptotes: y ! ±3x
"0, ±%10#"0, ±3#
"0, 0#
c ! %32 $ 12 ! %10
a ! 3, b ! 1,
–6 –4 –2 2 4 6
–6
6
x
yy2
9"
x2
1! 1
9.
Center:
Vertices:
Foci:
Asymptotes:
y ! ±ab
x ! ±59
x
"0, ±%106 #"0, ±5#
x
y
!6!9 6 9 12 15!3
!9!12!15
3
91215
"0, 0#
a ! 5, b ! 9, c ! %a2 $ b2 ! %106
y2
25"
x2
81! 1 10.
Center:
Vertices:
Foci:
Asymptotes: y ! ±13
x
"±2%10, 0#"±6, 0#
"0, 0#
c ! %36 $ 4 ! 2%10
a ! 6, b ! 2,
–12 12
–12
–8
–4
4
8
12
x
yx2
36"
y2
4! 1
11.
Center:
Vertices:
Foci:
Asymptotes: y ! "2 ±12
"x " 1#
"1 ± %5, "2#""1, "2#, "3, "2#
"1, "2#
a ! 2, b ! 1, c ! %5
1 2 3
–5
–4
1
2
3
x
y"x " 1#2
4"
"y $ 2#2
1! 1 12.
Center:
Vertices:
Foci:
Asymptotes:
y ! 2 ±512
"x $ 3#
""16, 2#, "10, 2#
""15, 2#, "9, 2#
a ! 12, b ! 5, c ! 13
""3, 2#5
10
15
!5 5!5
!10
!15
!20
x
y"x $ 3#2
144"
"y " 2#2
25! 1
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Section 9.3 Hyperbolas 797
13.
Center:
Vertices:
Foci:
Asymptotes:
x
y
!1!2 1 2 3 4!1
!2
!3
!5
y ! "5 ±23
"x " 1#
y ! k ±ab
"x " h#
&1, "5 ±%13
6 '&1, "5 ±
13': &1, "
163 ', &1, "
143 '
"1, "5#
a !13
, b !12
, c !%19
$14
!%13
6
"y $ 5#2
1(9"
"x " 1#2
1(4! 1 14.
Center:
Vertices:
Foci:
Asymptotes:
–3 –1
–1
1
2
3
x
y
y ! 1 ±1(21(4
"x $ 3# ! 1 ± 2"x $ 3#
&"3, 1 ±%54 '
&"3, 12', &"3,
32'
a !12
, b !14
, c !%14
$116
!%54
""3, 1#
"y " 1#2
1(4"
"x $ 3#2
1(16! 1
15. (a)
(b) Center:
Vertices:
Foci:
Asymptotes:
(c)
!2!4!5 2 4 5
!2!3!4!5
12345
x
y
y ! ±ba
x ! ±23
x
"±%13, 0#"±3, 0#
a ! 3, b ! 2, c ! %9 $ 4 ! %13
"0, 0#
x2
9"
y2
4! 1
4x2 " 9y2 ! 36 16. (a)
(b) Center:
Vertices:
Foci:
Asymptotes:
(c)
!4!6!8 4 6 8
!6
!8
6
8
4
x
y
y ! ±ba
x ! ±52
x
"±%29, 0#"±2, 0#
a ! 2, b ! 5, c ! %4 $ 25 ! %29
"0, 0#
x2
4"
y2
25! 1
25x2 " 4y2 ! 100
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Page 28
798 Chapter 9 Topics in Analytic Geometry
17. (a)
(b)
Center:
Vertices:
Foci:
! ±%63
x
Asymptotes: y ! ±%23
x
"±%5, 0#"±%3, 0#
"0, 0#
a ! %3, b ! %2, c ! %5
x2
3"
y2
2! 1
2x2 " 3y2 ! 6 (c) To use a graphing calculator, solve first for
y4 ! "%23
x
y3 !%23
x
y2 ! "%2x2 " 63
!3!4 3 4
!3
!2
!4
1
2
3
4
x
yy1 !%2x2 " 63
y2 !2x2 " 6
3
y.
) Asymptotes
) Hyperbola
18. (a)
(b)
Center:
Vertices:
Foci:
Asymptotes: y ! ±%3%6
x ! ±%22
x
"0, ±3#
"0, ±%3#"0, 0#
a ! %3, b ! %6, c ! 3
y 2
3"
x 2
6! 1
6y 2 " 3x 2 ! 18 (c)
x
y
!4 !3 !2 432!1
!3
!4
1
3
4
19. (a)
(b)
Center:
Vertices:
Foci:
Asymptotes: y ! "3 ± 3"x " 2#
"2 ± %10, "3#"1, "3#, "3, "3#
"2, "3#
a ! 1, b ! 3, c ! %10
"x " 2#2
1"
"y $ 3#2
9! 1
9"x2 " 4x $ 4# " "y2 $ 6y $ 9# ! "18 $ 36 " 9
9x2 " y2 " 36x " 6y $ 18 ! 0 (c)
–6 –4 –2 2 4 6 8
–8
–6
–4
2
x
y
20. (a)
x2
36"
"y " 2#2
4! 1
x2 " 9"y " 2#2 ! 36
x2 " 9" y2 " 4y $ 4# ! 72 " 36
x2 " 9y2 $ 36y " 72 ! 0 (b)
Center:
Vertices:
Foci:
Asymptotes: y ! 2 ±13
x
"±2%10, 2#"±6, 2#
"0, 2#
c ! %36 $ 4 ! 2%10
a ! 6, b ! 2, (c)
–8 –4 4 8
–12
–8
–4
4
8
12
x
y
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Section 9.3 Hyperbolas 799
21. (a)
(b) Degenerate hyperbola is two lines intersecting at""1, "3#.
y $ 3 ! ±13"x $ 1#
"x $ 1#2 " 9"y $ 3#2 ! 0
"x2 $ 2x $ 1# " 9"y2 $ 6y $ 9# ! 80 $ 1 " 81
x2 " 9y2 $ 2x " 54y " 80 ! 0 (c)
–4 –2 2
–6
–4
–2
2
4
x
y
22. (a)
(b) Degenerate hyperbola is two intersecting lines at "1, "2#.
y $ 2 ! ±14"x " 1#
16"y $ 2#2 " "x " 1# ! 0
16"y2 $ 4y $ 4# " "x2 " 2x $ 1# ! "63 $ 64 " 1
16y2 " x2 $ 2x $ 64y $ 63 ! 0 (c)
–1 1 2 3
–4
–3
–2
–1
x
y
23. (a)
(b)
Center:
Vertices:
Foci:
Asymptotes:
(c) To use a graphing calculator, solve for first.
y4 ! "3 "13
"x " 1#
y3 ! "3 $13
"x " 1#
y2 ! "3 "13%18 $ "x " 1#2
y1 ! "3 $13%18 $ "x " 1#2
y ! "3 ± %18 $ "x " 1#2
9
9"y $ 3#2 ! 18 $ "x " 1#2
x
y
2
!6
!8
!10
2
4
y
y ! "3 ±13
"x " 1#
"1, "3 ± 2%5 #"1, "3 ± %2 #
"1, "3#
a ! %2, b ! 3%2, c ! 2%5
"y $ 3#2
2"
"x " 1#2
18! 1
9"y2 $ 6y $ 9# " "x2 " 2x $ 1# ! "62 " 1 $ 81
9y2 " x2 $ 2x $ 54y $ 62 ! 0
) Asymptotes
) Hyperbola
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800 Chapter 9 Topics in Analytic Geometry
24. (a)
(b)
Center:
Vertices:
Foci:
Asymptotes: y ! 5 ± 3"x $ 3#
&"3 ±%10
3, 5'
&"3 ±13
, 5'""3, 5#
a !13
, b ! 1, c !%10
3
"x $ 3#2
1(9"
"y " 5#2
1! 1
9"x2 $ 6x $ 9# " "y2 " 10y $ 25 # ! "55 $ 81 " 25
9x2 " y2 $ 54x $ 10y $ 55 ! 0 (c)
x
y
!2!3!4!6!7 1
2
4
6
8
10
14
28. Vertices:
Asymptotes:
Center:
y2
9" x2 ! 1
"y " k#2
a2 ""x " h#2
b2 ! 1
"0, 0# ! "h, k#
y ! ±3x ! ab
! 3, b ! 1
"0, ±3# ! a ! 3 29. Foci:
Asymptotes:
Center:
17y2
1024"
17x2
64! 1
y2
1024(17"
x2
64(17! 1
"y " k#2
a2 ""x " h#2
b2 ! 1
6417
! b2 ! a2 !102417
c2 ! a2 $ b2 ! 64 ! 16b2 $ b2
"0, 0# ! "h, k#
y ! ±4x ! ab
! 4 ! a ! 4b
"0, ±8# ! c ! 8
25. Vertices:
Foci:
Center:
y2
4"
x2
12! 1
"y " k#2
a2 ""x " h#2
b2 ! 1
"0, 0# ! "h, k#
b2 ! c2 " a2 ! 16 " 4 ! 12
"0, ±4# ! c ! 4
"0, ±2# ! a ! 2 26. Vertices:
Foci:
x2
9"
y2
27! 1
x2
a2 "y2
b2 ! 1
b2 ! c2 " a2 ! 36 " 9 ! 27
"±6, 0# ! c ! 6
"±3, 0# ! a ! 3 27. Vertices:
Asymptotes:
Center:
x2
1"
y2
25! 1
"0, 0#
! b ! 5
y ! ±5x ! ba
! 5
"±1, 0# ! a ! 1
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Section 9.3 Hyperbolas 801
30. Foci:
Asymptotes:
x2
64"
y2
36! 1
x2
a2 "y2
b2 ! 1
b ! 3"2# ! 6a ! 4"2# ! 8,
2 ! m
100 ! 25m2
c2 ! a2 $ b2 ! 100 ! "3m#2 $ "4m#2
y ! ±34
x ! ba
!3m4m
"±10, 0# ! c ! 10 31. Vertices:
Foci:
Center:
"x " 4#2
4"
y2
12! 1
"x " h#2
a2 ""y " k#2
b2 ! 1
"4, 0# ! "h, k#
b2 ! c2 " a2 ! 16 " 4 ! 12
"0, 0#, "8, 0# ! c ! 4
"2, 0#, "6, 0# ! a ! 2
32. Vertices:
Center:
Foci:
y2
9"
"x " 2#2
16! 1
"y " k#2
a2 ""x " h#2
b2 ! 1
b2 ! c2 " a2 ! 25 " 9 ! 16
"2, "5# ! c ! 5"2, 5#,
"2, 0#
"2, "3# ! a ! 3"2, 3#, 33. Vertices:
Foci:
Center:
"y " 5#2
16"
"x " 4#2
9! 1
"y " k#2
a2 ""x " h#2
b2 ! 1
"4, 5# ! "h, k#
b2 ! c2 " a2 ! 25 " 16 ! 9
"4, 0#, "4, 10# ! c ! 5
"4, 1#, "4, 9# ! a ! 4
34. Vertices:
Center:
Foci:
x2
4"
"y " 1#2
5! 1
"x " h#2
a2 ""y " k#2
b2 ! 1
b2 ! c2 " a2 ! 9 " 4 ! 5
"3, 1# ! c ! 3""3, 1#,
"0, 1#
"2, 1# ! a ! 2""2, 1#, 35. Vertices:
Solution point:
Center:
y2
9"
"x " 2#2
9(4! 1
!9""2#2
25 " 9!
3616
!94
b2 !9"x " 2#2
y2 " 9
y2
9"
"x " 2#2
b2 ! 1 !
"y " k#2
a2 ""x " h#2
b2 ! 1
"2, 0# ! "h, k#
"0, 5#
"2, 3#, "2, "3# ! a ! 3
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Page 32
802 Chapter 9 Topics in Analytic Geometry
36. Center:
Solution point:
x2
4"
"y " 1#2
12(7! 1
b2 !3621
!127
9b2 !
214
254
"9b2 ! 1
"5, 4#
x2
4"
"y " 1#2
b2 ! 1
"0, 1#, a ! 2 37. Vertices:
Center:
Passes through
"y " 2#2
4"
x2
4! 1
b2 ! 4 ! b ! 2
94
" 1 !5b2
""1 " 2#2
4"
5b2 ! 1
"%5, "1#
"y " 2#2
4"
x2
b2 ! 1
"0, 2#, a ! 2
"0, 4#, "0, 0#
41. Vertices:
Asymptotes:
Center:
"x " 3#2
9"
"y " 2#2
4! 1
"x " h#2
a2 ""y " k#2
b2 ! 1
"3, 2# ! "h, k#
ba
!23
! b ! 2
y !23
x, y ! 4 "23
x
"0, 2#, "6, 2# ! a ! 3 42. Vertices:
Asymptotes:
Center:
"y " 2#2
4"
"x " 3#2
9! 1
"y " k#2
a2 ""x " h#2
b2 ! 1
"3, 2# ! "h, k#
ab
!23
! b ! 3
y !23
x, y ! 4 "23
x
(3, 0#, "3, 4# ! a ! 2
38. Center:
Solution point:
y2
4"
"x " 1#2
4! 1
1b2 !
14
! b ! 2
54
"1b2 ! 1
"0, %5#
y2
4"
"x " 1#2
b2 ! 1
"1, 0#, a ! 2 39. Vertices:
Center:
Asymptotes:
"x " 2#2
1"
"y " 2#2
1! 1
ba
! 1 ! b ! 1
y ! x, y ! 4 " x
"2, 2#
"1, 2#, "3, 2# ! a ! 1
40. Center:
Asymptotes:
"y $ 3#2
9"
"x " 3#2
9! 1
1 !ab
!3b
! b ! 3
y ! x " 6, y ! "x
"3, "3#, a ! 3
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Page 33
Section 9.3 Hyperbolas 803
43. Friend’s location
Your location
Location of lightning strike
x2
98,010,000"
y2
13,503,600
b2 ! c2 " a2 ! 13,503,600
c ! 10,560, a !19,800
2! 9900 ! a2 ! 98,010,000
x2
a2 "y2
b2 ! 1
"1100#"18# ! 19,800
P"x, y#:
"10,560, 0#F2:
!20,000 20,000
!10,000
10,000
x
y
P
F
(10,560, 0)(!10,560, 0)
F1 2Friend You
""10,560, 0#F1:
44. The explosion occurred on the vertical line throughand
Hence,
The explosion occurred on the hyperbola
Letting
"3300, "2750#
y2 ! b2&x2
a2 " 1' ! "33002 " 22002#&33002
22002 " 1' ! y ! "2750.
x ! 3300,
x2
a2 "y2
b2 ! 1.
b2 ! c2 " a2.
c ! 3300
a ! 2200
2a ! 4400
d2 " d1 ! 4"1100# ! 4400
"3300, 0#."3300, 1100# (3300, 1100)
(3300, 0)( 3300, 0)!
d1d2
1000
2000
3000
4000
ax
y
!4000
!4000
45. (a)
is on the curve, so
x2
1"
y2
27! 1, "9 " y " 9
! b2 !813
! b ! 3%3.
41
"81b2 ! 1 ! 81
b2 ! 3
a ! 1; "2, 9#
x2
a2 "y2
b2 ! 1 (b) Because each unit is foot, 4 inches is of a unit. The base is 9 units from the origin, so
When
So the width is units, or22.68 inches, or 1.88998 feet.
2x $ 3.779956
x2 ! 1 $"25(3#2
27 ! x $ 1.88998.
y !253
,
y ! 9 "23
! 813
.
23
12
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Page 34
804 Chapter 9 Topics in Analytic Geometry
46. Foci:
Center:
(a)
(b) 150 " 93 ! 57 miles
x $ 110.3 miles
x2 ! 932&1 $752
13,851' $ 12,161.43
x2
932 "y2
13,851! 1
b2 ! c2 " a2 ! 1502 " 932 ! 13,851
! 186 ! 2a ! 186 ! a ! 93
d2 " d1 ! "186,000#"0.001#
d1d2
15075
75
150
(150, 0)x
y
!75!150
(!150, 0)
(x, 75)
"0, 0#
"±150, 0# ! c ! 150
(c) Bay to Station 1: 30 miles
Bay to Station 2: 270 miles
(d) In this case,
and The hyperbola is
For and
Position: "144.2, 60#
x $ 144.2.y ! 60, x2 ! 20,800
x2
1202 "y2
902 ! 1.
b2 ! c2 " a2 ! 8100.
d2 " d1 ! 186,000"0.00129# $ 239.94 ! a $ 120
"270 " 30#186,000
$ 0.00129 second
47. Center:
Focus:
Since and we choose The vertex is approximate at [Note: By the Quadratic Formula, the exact value of is ]a ! 12"%5 " 1#.a
"14.83, 0#.a ! 14.83.c ! 24,a < c
a $ ±38.83 or a $ ±14.83
a4 " 1728a2 $ 331,776 ! 0
576"576 " a2# " 576a2 ! a2"576 " a2#
576a2 "
576576 " a2 ! 1
242
a2 "242
576 " a2 ! 1
x2
a2 "y2
576 " a2 ! 1
b2 ! c2 " a2 ! 242 " a2 ! 576 " a2
"24, 0#
"0, 0#
48.
The camera is units from the mirror.5 $ %41
a ! 5, b ! 4, c ! %25 $ 16 ! %41
x2
25"
y2
16! 1 49.
EllipseAC ! 36 > 0,
A ! 9, C ! 4
9x2 $ 4y2 " 18x $ 16y " 119 ! 0 ©H
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Page 35
Section 9.3 Hyperbolas 805
50.
CircleA ! C ! 1,
x2 $ y2 " 4x " 6y " 23 ! 0 51.
HyperbolaAC ! 16""9# < 0,
A ! 16, C ! "9
16x2 " 9y2 $ 32x $ 54y " 209 ! 0
52.
ParabolaAC ! 0,
A ! 1, C ! 0
x2 $ 4x " 8y $ 20 ! 0 53.
ParabolaAC ! 0,
C ! 1, A ! 0
y2 $ 12x $ 4y $ 28 ! 0
54.
EllipseAC ! 100 > 0,
A ! 4, C ! 25
4x2 $ 25y2 $ 16x $ 250y $ 541 ! 0 55.
CircleA ! C ! 1,
x2 $ y2 $ 2x " 6y ! 0
56.
HyperbolaAC < 0,
A ! "1, C ! 1
y2 " x2 $ 2x " 6y " 8 ! 0 57.
AC ! 0 ! Parabola
E ! "2, F ! 7A ! 1, C ! 0, D ! "6,
x2 " 6x " 2y $ 7 ! 0
58.
AC ! 9"4# ! 36 > 0 ! Ellipse
A ! 9, C ! 4
9x2 $ 4y2 " 90x $ 8y $ 228 ! 0 59. True. e !ca
!%a2 $ b2
a
60. False. because it is in the denominator.b # 0
61. False. For example,
is the graph of two intersecting lines.
"x " 1#2 " " y " 1#2 ! 0
x2 " y2 " 2x $ 2y ! 0
62. True. The asymptotes are
If they intersect at right angles, then
ba
!"1
""b(a# !ab
! a ! b.
y ! ±ba
x.
63. Let be such that the difference of the distances from and is (again only deriving one of the forms).
Let Then a2b2 ! b2x2 " a2y2 ! 1 !x2
a2 "y2
b2.b2 ! c2 " a2.
a2"c2 " a2# ! "c2 " a2#x2 " a2y2
a2"x2 " 2cx $ c2 $ y2# ! c2x2 " 2a2cx $ a4
a%"x " c#2 $ y2 ! cx " a2
4a%"x " c#2 $ y2 ! 4cx " 4a2
4a2 $ 4a%"x " c#2 $ y2 $ "x " c#2 $ y2 ! "x $ c#2 $ y2
2a $ %"x " c#2 $ y2 ! %"x $ c#2 $ y2
2a ! *%"x $ c#2 $ y2 " %"x " c# $ y2*2a
""c, 0#"c, 0#"x, y#
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Page 36
806 Chapter 9 Topics in Analytic Geometry
64. Answers will vary. See Example 3. 65.
At the point
*d2 " d1* ! *"a $ c# " "c " a#* ! 2a.
"a, 0#,*d2 " d1* ! constant by definition of hyperbola
66. Center:
Horizontal transverse axis
Foci at and
"x " 6#2
9"
" y " 2#2
7! 1
b2 ! c2 " a2 ! 16 " 9 ! 7
"c $ a# " "c " a# ! 6 ! a ! 3
"10, 2# ! c ! 4."2, 2#
"6, 2#
67. At the point the difference of the distances to the foci is Let be a point on the hyperbola.
Thus, as desired.c2 " a2 ! b2,
1 !x2
a2 "y2
c2 " a2
a2"c2 " a2# ! "c2 " a2#x2 " a2y2
a2"x2 " 2cx $ c2 $ y2# ! c2x2 " 2a2cx $ a4
a%"x " c#2 $ y2 ! cx " a2
4a%"x " c#2 $ y2 ! 4cx " 4a2
4a2 $ 4a%"x " c#2 $ y2 $ "x " c#2 $ y2 ! "x $ c#2 $ y2
2a $ %"x " c#2 $ y2 ! %"x $ c#2 $ y2
2a ! %"x $ c#2 $ y2 " %"x " c#2 $ y2
"x, y#"c $ a# " "c " a# ! 2a."±c, 0#"a, 0#,
68. If then by completing the square you obtain a circle.
If and then is a parabola (complete the square).Same for and
If then both and are positive (or both negative). By completing the squareyou obtain an ellipse.
If then and have opposite signs. You obtain a hyperbola.CAAC < 0,
CAAC > 0,
C ! 0.A # 0Cy2 $ Dx $ Ey $ F ! 0C # 0,A ! 0
A ! C # 0,
69. "x3 " 3x2# " "6 " 2x " 4x2# ! x3 $ x2 $ 2x " 6 70.
! 3x2 $ 232 x " 2
"3x " 12#"x $ 4# ! 3x2 $ 12x " 1
2x " 2
71.
x3 " 3x $ 4x $ 2
! x2 " 2x $ 1 $2
x $ 2
"2 1
1
0"2
"2
"34
1
4"2
2
72.
! x2 $ 2xy $ y2 $ 6x $ 6y $ 9
+"x $ y# $ 3,2 ! "x $ y#2 $ 6"x $ y# $ 9
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Page 37
Section 9.4 Rotation and Systems of Quadratic Equations 807
Section 9.4 Rotation and Systems of Quadratic Equations
! The general second-degree equation can be rewritten by rotating the coordinate axes through the angle
where
!
! The graph of the nondegenerate equation is:
(a) An ellipse or circle if
(b) A parabola if
(c) A hyperbola if B2 ! 4AC > 0.
B2 ! 4AC " 0.
B2 ! 4AC < 0.
Ax2 # Bxy # Cy2 # Dx # Ey # F " 0
y " x$ sin % # y$ cos %x " x$ cos % ! y$sin %
cot 2% " !A ! C"#B.%A$!x$ "2 # C$!y$ "2 # D$x$ # E$y$ # F$ " 0
Ax2 # Bxy # Cy2 # Dx # Ey # F " 0
Vocabulary Check1. rotation, axes 2. invariant under rotation 3. discriminant
1. Point:
Thus, !x$, y$ " " !3, 0".
3 " x$0 " y$
3 " x$ sin 90& # y$ cos 90&0 " x$ cos 90& ! y$ sin 90&
y " x$ sin % # y$ cos %x " x$ cos % ! y$ sin %
!0, 3"% " 90&;
2. Point:
Adding,
Subtracting,
Thus, !x$, y$ " " !3$2, 0".$2y$ " 0 ! y$ " 0.
6 " $2x$ ! x$ "6$2
" 3$2.
3 "$22
x$ #$22
y$3 "$22
x$ !$22
y$
3 " x$ sin 45& # y$ cos 45&3 " x$ cos 45& ! y$ sin 45&
y " x$ sin % # y$ cos %x " x$ cos % ! y$ sin %
!3, 3"% " 45&;
73. x3 ! 16x " x!x2 ! 16" " x!x ! 4"!x # 4" 74. x2 # 14x # 49 " !x # 7"2
75.
" 2x!x ! 6"2
2x3 ! 24x2 # 72x " 2x!x2 ! 12x # 36" 76.
" x!3x # 2"!2x ! 5"
6x3 ! 11x2 ! 10x " x!6x2 ! 11x ! 10"
77.
" 2!2x # 3"!4x2 ! 6x # 9"
16x3 # 54 " 2!8x3 # 27" 78.
" !4 ! x"!x # i"!x ! i"
" !4 ! x"!x2 # 1"
4 ! x # 4x2 ! x3 " !4 ! x" # x2!4 ! x"
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Page 38
808 Chapter 9 Topics in Analytic Geometry
3.
Hyperbola !y$ "2
2!
!x$ "2
2" 1,
%x$ ! y$
$2 &%x$ # y$
$2 & # 1 " 0
xy # 1 " 0
"x$ # y$
$2 " x$%$2
2 & # y$%$22 & y " x$ sin
'
4# y$ cos
'
4
"x$ ! y$
$2 " x$%$2
2 & ! y$%$22 & x " x$ cos
'
4! y$ sin
'
4
cot 2% "A ! C
B" 0 ! 2% "
'
2 ! % "
'
4
A " 0, B " 1, C " 0
!4 !3 !2 4
!4
!3
!2
4y " x "
x
yxy # 1 " 0
4.
, Hyperbola !x$ "2
4!
!y$ "2
4" 1
!x$ "2 ! !y$ "2
2" 2
%x$ ! y$
$2 &%x$ # y$
$2 & ! 2 " 0
xy ! 2 " 0
y " x$ sin '4
# y$ cos '4
" x$%$22 & # y$%$2
2 & "x$ # y$
$2
x " x$ cos '4
! y$ sin '4
" x$%$22 & ! y$%$2
2 & "x$ ! y$
$2
cot 2% "A ! C
B" 0 ! 2% "
'
2 ! % "
'
4 468
10
64 8 10
!8!10
x"y"
x
yxy ! 2 " 0, A " 0, B " 1, C " 0
5.
"$22
!x$ ! y$ "
" x$%$22 & ! y$%$2
2 & x " x$ cos
'4
! y$ sin '4
cot 2% "A ! C
B" 0 ! 2% "
'2
! % "'4
A " 1, B " !4, C " 1!4!6!8 4 6 8
!6
!8
4
6
8
x
y
y " x "
x2 ! 4xy # y2 # 1 " 0
y " x$ sin '4
# y$ cos '4
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Page 39
Section 9.4 Rotation and Systems of Quadratic Equations 809
5. —CONTINUED—
, Hyperbola !x$ "2 !!y$ "2
1#3" 1
!!x$ "2 # 3!y$ "2 " !1
12
!x$ "2 ! x$y$ #12
! y$ "2 ! 2'!x$ "2 ! !y$ "2( #12
!x$ "2 # x$y$ #12
!y$ "2 # 1 " 0
)$22
!x$ ! y$ "*2
! 4)$22
!x$ ! y$ "$22
!x$ # y$ "* # )$22
!x$ # y$ "*2
# 1 " 0
x2 ! 4xy # y2 # 1 " 0
6.
, Hyperbola %y$ #
3$22 &2
10!%x$ !
$22 &2
10" 1
%x$ !$22 &2
! %y$ #3$2
2 &2
" !10
)!x$ "2 ! $2x$ # %$22 &2* ! )!y$ "2 # 3$2y$ # %3$2
2 &2* " !6 # %$22 &2
! %3$22 &2
!x$ "2
2!
!y$ "2
2#
x$$2
!y$
$2!
2x$
$2!
2y$
$2# 3 " 0
%x$ ! y$
$2 &%x$ # y$
$2 & # %x$ ! y$
$2 & ! 2%x$ # y$
$2 & # 3 " 0
xy # x ! 2y # 3 " 0
"x$ ! y$
$2 "
x$ # y$
$2
" x$%$22 & ! y$%$2
2 & " x$%$22 & # y$%$2
2 & x " x$ cos
'
4! y$ sin
'
4 y " x$ sin
'
4# y$ cos
'
4
cot 2% "A ! C
B" 0 ! 2% "
'
2 ! % "
'
4
A " 0, B " 1, C " 0
x
x"y"
!4 4!6!8 6
4
6
8
!4
!6
!8
yxy # x ! 2y # 3 " 0
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Page 40
810 Chapter 9 Topics in Analytic Geometry
7.
!x$ ! 3$2"2
16!
!y$ ! $2 "2
16" 1, Hyperbola
!x$ ! 3$2 "2! !y$ ! $2 "2
" 16
'!x$ "2 ! 6$2x$ # !3$2 "2( ! '!y$ "2 ! 2$2y$ # !$2 "2( " 0 # !3$2 "2! !$2 "2
!x$ "2
2!
!y$ "2
2! $2x$ ! $2y$ ! 2$2x$ # 2$2y$ " 0
%x$ ! y$
$2 &%x$ # y$
$2 & ! 2%x$ # y$
$2 & ! 4%x$ ! y$
$2 & " 0
xy ! 2y ! 4x " 0
"x$ ! y$
$2
" x$%$22 & ! y$%$2
2 & x " x$ cos
'
4! y$ sin
'
4
cot 2% "A ! C
B" 0 ! 2% "
'
2 ! % "
'
4
A " 0, B " 1, C " 0
x
x"
y "4
6
8
!4
!4 2 4 6 8
yxy ! 2y ! 4x " 0
"x$ # y$
$2
" x$%$22 & # y$%$2
2 & y " x$ sin
'
4# y$ cos
'
4
8.
—CONTINUED—
"x$ ! 3y$
$10 "
3x$ # y$
$10
" x$% 1$10& ! y$% 3
$10& " x$% 3$10& # y$% 1
$10& x " x$ cos % ! y$ sin % y " x$ sin % # y$ cos %
cos % "$1 # cos 2%
2"$1 # !!4#5"
2"
1$10
sin % "$1 ! cos 2%
2"$1 ! !!4#5"
2"
3$10
cos 2% " !45
cot 2% "( ! C
B" !
43
! % + 71.57&
A " 2, B " !3, C " !2
x
x"
y" 2
4
!4
!4 !2 4
y2x2 ! 3xy ! 2y2 # 10 " 0
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Page 41
Section 9.4 Rotation and Systems of Quadratic Equations 811
8. —CONTINUED—
, Hyperbola !x$ "2
4!
!y$ "2
4" 1
!52
!x$ "2 #52
!y$ "2 " !10
!x$ "2
5!
6x$y$
5#
9!y$ "2
5!
9!x$ "2
10#
24x$y$
10#
9!y$ "2
10!
9!x$ "2
5!
6x$y$
5!
!y$ "2
5# 10 " 0
2%x$ ! 3y$
$10 &2
! 3%x$ ! 3y$
$10 &%3x$ # y$
$10 & ! 2%3x$ # y$
$10 &2
# 10 " 0
2x2 ! 3xy ! 2y2 # 10 " 0
9.
!x$ "2
6#
!y$ "2
3#2" 1, Ellipse
2!x$ "2 # 8!y$ "2 " 12
52
!x$ "2 ! 5x$y$ #52
!y$ "2 ! 3!x$ "2 # 3!y$ "2 #52
!x$ "2 # 5x$y$ #52
!y$ "2 " 12
5)$22
!x$ ! y$ "*2
! 6)$22
!x$ ! y$ " $22
!x$ # y$ "* # 5)$22
!x$ # y$ "*2
" 12
5x2 ! 6xy # 5y2 ! 12 " 0
y " x$ sin '4
# y$ cos '4
"$22
!x$ # y$ "
x " x$ cos '4
! y$ sin '4
"$22
!x$ ! y$ "
% "'4
!2% "'2
!cot 2% "A ! C
B" 0
A " 5, B " !6, C " 5 x"y"
2
2
3
!3
!3!4
!4
4
3 4x
y5x2 ! 6xy # 5y2 ! 12 " 0
10.
—CONTINUED—
"$3x$ ! y$
2 "
x$ # $3y$
2
" x$%$32 & ! y$%1
2& " x$%12& # y$%$3
2 &x " x$ cos
'
6! y$ sin
'
6 y " x$ sin
'
6# y$ cos
'
6
cot 2% "A ! C
B"
1$3
! 2% "'
3 ! % "
'
6
A " 13, B " 6$3, C " 7
!3 !2 2 3
!3
!2
3
x
y "
x "
y13x2 # 6$3xy # 7y2 ! 16 " 0
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Page 42
812 Chapter 9 Topics in Analytic Geometry
10. —CONTINUED—
, Ellipse !x$ "2
1#
!y$ "2
4" 1
16!x$ "2 # 4!y$ "2 " 16
!18!y$ "2
4#
7!x$ "2
4#
7$3x$y$
2#
21!y$ "2
4! 16 " 0
39!x$ "2
4!
13$3x$y$
2#
13!y$ "2
4#
18!x$ "2
4#
18$3x$y$
4!
6$3x$y$
4
13%$3x$ ! y$
2 &2
# 6$3%$3x$ ! y$
2 &%x$ # $3y$
2 & # 7%x$ # $3y$
2 &2
! 16 " 0
13x2 # 6$3xy # 7y2 ! 16 " 0
11.
Parabola x$ " !!y$ "2,
4(y$ )2# 4x$ " 0
# x$ ! $3y$ # 3x$ # $3y$ " 0
3(x$ ) 2
4!
6$3x$y$
4#
9(y$ ) 2
4!
6(x$ )2
4#
4$3x$y$
4#
6(y$ )2
4#
3(x$ ) 2
4#
2$3x$y$
4#
(y$ ) 2
4
# 2$3%$3x$ # y$
2 & " 0
3%x$ ! $3y$
2 &2
! 2$3%x$ ! $3y$
2 &%$3x$ # y$
2 & # %$3x$ # y$
2 &2
# 2%x$ ! $3y$
2 &
3x2 ! 2$3xy # y2 # 2x # 2$3y " 0
y " x$ sin % # y$ cos % "$3x$ # y$
2
" x$%12& ! y$%$3
2 & "x$ ! $3y$
2
x " x$ cos 60& ! y$ sin 60&
cot 2% "A ! C
B" !
1$3
! % " 60&
A " 3, B " !2$3, C " 1x
x"y"
2!4!6
2
!2
!4
y3x2 ! 2$3xy # y2 # 2x # 2$3y " 0
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Page 43
Section 9.4 Rotation and Systems of Quadratic Equations 813
12.
Parabola
!y$ "2 " 4!x$ ! 1"
25!y$ "2 ! 100x$ # 100 " 0
#81!y$ "2
25! 36x$ # 48y$ ! 64x$ ! 48y$ # 100 " 0
144!x$ "2
25!
384x$y$
25#
256!y$ "2
25!
288!x$ "2
25#
168x$y$
25#
288!y$ "2
25#
144!x$ "2
25#
216x$y$
25
16%3x$ ! 4y$
5 &2
! 24%3x$ ! 4y$
5 &%4x$ # 3y$
5 &# 9%4x$ # 3y$
5 &2
! 60%3x$ ! 4y$
5 &! 80%4x$ # 3y$
5 &# 100 " 0
16x2 ! 24xy # 9y2 ! 60x ! 80y # 100 " 0
y " x$ sin % # y$ cos % " x$%45& # y$%3
5& "4x$ # 3y$
5
x " x$ cos % ! y$ sin % " x$%35& ! y$%4
5& "3x$ ! 4y$
5
cos % "$1 # cos 2%
2"$1 # !!7#25"
2"
35
sin % "$1 ! cos 2%
2"$1 ! !!7#25"
2"
45
cos 2% " !725
cot 2% "( ! C
B" !
724
! % + 53.13&
A " 16, B " !24, C " 9
x1 2 3 4 5 6
1
x"
y"
y16x2 ! 24xy # 9y2 ! 60x ! 80y # 100 " 0
13.
—CONTINUED—
"3x$ ! 4y$
5
" x$%35& ! y$%4
5& x " x$ cos % ! y$ sin %
cos % "$1 # cos 2%
2"$1 # !!7#25"
2"
35
sin % "$1 ! cos 2%
2"$1 ! !!7#25"
2"
45
cos 2% " !725
cot 2% "A ! C
B" !
724
! % + 53.13&
A " 9, B " 24, C " 16
x
x "y"
!4 42
!2
2
4
6
y9x2 # 24xy # 16y2 # 90x ! 130y " 0
"4x$ # 3y$
5
" x$%45& # y$%3
5& y " x$ sin % # y$ cos %
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Page 44
814 Chapter 9 Topics in Analytic Geometry
13. —CONTINUED—
Parabola !x$ ! 1"2 " 6%y$ #16&,
!x$ "2 ! 2x$ # 1 " 6y$ # 1
25!x$ "2 ! 50x$ ! 150y$ " 0
#144!y$"2
25# 54x$ ! 72y$ ! 104x$ ! 78y$ " 0
81!x$ "2
25!
216x$y$
25#
144!y$ "2
25#
288!x$ "2
25!
168x$y$
25!
288!y$ "2
25#
256!x$ "2
25#
384x$y$
25
! 130%4x$ # 3y$5 & " 0
9%3x$ ! 4y$
5 &2
# 24%3x$ ! 4y$
5 &%4x$ # 3y$
5 & # 16%4x$ # 3y$
5 &2
# 90%3x$ ! 4y$
5 &
9x2 # 24xy # 16y2 # 90x ! 130y " 0
14.
Parabola
!x$ "2 " 4y$,
25!x$ "2 ! 100y$ " 0
#144!y$ "2
25# 48x$ ! 64y$ ! 48x$ ! 36y$ " 0
81!x$ "2
25!
216x$y$
25#
144!y$ "2
25#
288!x$ "2
25!
168x$y$
25!
288!y$ "2
25#
256!x$ "2
25#
384x$y$
25
9%3x$ ! 4y$
5 &2
# 24%3x$ ! 4y$
5 &%4x$ # 3y$
5 & # 16%4x$ # 3y$
5 &2
# 80%3x$ ! 4y$
5 & ! 60%4x$ # 3y$
5 & " 0
9x2 # 24xy # 16y2 # 80x ! 60y " 0
y " x$ sin % # y$ cos % " x$%45& # y$%3
5& "4x$ # 3y$
5
x " x$ cos % ! y$ sin % " x$%35& ! y$%4
5& "3x$ ! 4y$
5
x " x$ cos % ! y$ sin % y " x$ sin % # y$ cos %
cos % "$1 # cos 2%
2"$1 # !!7#25"
2"
35
sin % "$1 ! cos 2%
2"$1 ! !!7#25"
2"
45
cos 2% " !725
cot 2% "( ! C
B" !
724
! % + 53.13&
A " 9, B " 24, C " 16
x
y
!1
1
2
3
x "
y "
!3 !2 !1 1
9x2 # 24xy # 16y2 # 80x ! 60y " 0
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Page 45
Section 9.4 Rotation and Systems of Quadratic Equations 815
15.
Solve for in terms of
y " !32
x ±$80 # 5x2
2
%y #32
x&2" 20 #
5x2
4"
80 # 5x2
4
y2 # 3xy #9x2
4" 20 ! x2 #
9x2
4
y2 # 3xy " 20 ! x2
x:y
cot 2% "A ! C
B"
1 ! 13
" 0 ! % "'4
" 45&
x2 # 3xy # y2 " 20
Graph and
!8
!12 12
8
y2 " !3x2
!$80 # 5x2
2.y1 " !
3x2
#$80 # 5x2
2
16.
To graph conic with a graphing calculator, we needto solve for in terms of
Graph and
!6
!9 9
6
y2 " x !$4 #x2
2.
y1 " x #$4 #x2
2
y " x ± $4 #x2
2
y ! x " ±$4 #x2
2
!y ! x"2 " 4 #x2
2
y2 ! 2xy # x2 " 4 !x2
2# x2
x2 ! 4xy # 2y2 " 8
x.y
% + 37.98&
2% + 75.96&
tan 2% " 4
1
tan 2%"
14
cot 2% "A ! C
B"
1 ! 2!4
"14
A " 1, B " !4, C " 2
x2 ! 4xy # 2y2 " 8 17.
Solve for in terms of by completing the square.
Graph and
!6
!9 9
6
y2 "16x ! 5$15x2 ! 21
7.
y1 "16x # 5$15x2 ! 21
7
y "16x ± 5$15x2 ! 21
7
y "167
x ± $375x2 ! 52549
%y !167
x&2
"375x2 ! 525
49
y2 !327
xy #25649
x2 "11949
x2 !52549
#25649
x2
y2 !327
xy "177
x2 !757
!7y2 # 32xy " !17x2 # 75
xy
cot 2% "A ! C
B"
17 # 732
"2432
"34
! % + 26.57&
17x2 # 32xy ! 7y2 " 75
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Page 46
816 Chapter 9 Topics in Analytic Geometry
18.
Solve for in terms of by completing the square:
Graph and
y2 "!18x ! $1300 ! 676x2
25.
y1 "!18x # $1300 ! 676x2
25
y "!18x ± $1300 ! 676x2
25
y #1825
x " ±$1300 ! 676x2
625
%y #1825
x&2
"1300 ! 676x2
625
y2 #3625
xy #324625
x2 "5225
!4025
x2 #324625
x2
y2 #3625
xy "5225
!4025
x2
25y2 # 36xy " 52 ! 40x2
xy
% + 33.69&
2% + 67.38&
tan 2% "125
!2
!3 3
2 1
tan 2%"
512
cot 2% "A ! C
B"
40 ! 2536
"512
A " 40, B " 36, C " 25
40x2 # 36xy # 25y2 " 52 19.
Solve for in terms of
Graph and
!6
!9 9
6
y2 " !3x !$20x2 # 25
2.
y1 " !3x #$20x2 # 25
2
y " !3x ±$20x2 # 25
2
!y # 3x"2 " 5x2 #254
"20x2 # 25
4
y2 # 6xy # 9x2 " !4x2 #254
# 9x2
y2 # 6xy " !4x2 #254
8y2 # 48xy " !32x2 # 50
x:y
cot 2% "A ! C
B"
32 ! 848
"12
! % + 31.72&
32x2 # 48xy # 8y2 " 50
20.
—CONTINUED—
% + 33.69&
2% + 67.38&
tan 2% "125
1
tan 2%"
512
cot 2% "A ! C
B"
4 ! 9!12
"512
A " 4, B " !12, C " 9
4x2 ! 12xy # 9y2 # !4$13 ! 12"x ! !6$13 # 8"y " 91
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Page 47
Section 9.4 Rotation and Systems of Quadratic Equations 817
20. —CONTINUED—
Solve for in terms of with the Quadratic Formula:
Graph and
y2 "12x # 6$13 # 8 ! $624x # 3808 # 96$13
18.
y1 "12x # 6$13 # 8 # $624x # 3808 # 96$13
18!6
!9 27
18 "!12x # 6$13 # 8" ± $624x # 3808 # 96$13
18
y "!12x # 6$13 # 8" ± $!12x # 6$13 # 8"2 ! 4!9"!4x2 # 4$13x ! 12x ! 91"
18
y "!b ± $b2 ! 4ac
2a
a " 9, b " !!12x # 6$13 # 8", c " 4x2 # 4$13x ! 12x ! 91
9y2 ! !12x # 6$13 # 8"y # !4x2 # 4$13x ! 12x ! 91" " 0
4x2 ! 12xy # 9y2 # !4$13 ! 12"x ! !6$13 # 8"y " 91
xy
21.
The graph is a hyperbola.
Matches graph (e).
cot 2% "A ! C
B" 0 ! % " 45&
B2 ! 4AC " 1 !
xy # 4 " 0 22.
The graph is a line. Matches graph (b).
y " !x
x # y " 0
!x # y"2 " 0
x2 # 2xy # y2 " 0
23.
Matches graph (f).
cot 2% "A ! C
B" !
43
! % + !18.43&
" 25 ! The graph is a hyperbola.
B2 ! 4AC " !3"2 ! 4!!2"!2"
!2x2 # 3xy # 2y2 # 3 " 0 24.
The graph is an ellipse or circle.
Matches graph (a).
cot 2% "A ! C
B"
1 ! 3!1
" 2 ! % + 13.28&
B2 ! 4AC " !!1"2 ! 4!1"!3" " !11
A " 1, B " !1, C " 3
x2 ! xy # 3y2 ! 5 " 0
25.
Matches graph (d).
cot 2% "A ! C
B" 1 ! % " 22.5&
" !8 ! The graph is an ellipse or circle.
B2 ! 4AC " !2"2 ! 4!3"!1"
3x2 # 2xy # y2 ! 10 " 0 26.
The graph is a parabola.
Matches graph (c).
cot 2% "A ! C
B"
1 ! 4!4
"34
! % + 26.57&
B2 ! 4AC " !!4"2 ! 4!1"!4" " 0
A " 1, B " !4, C " 4
x2 ! 4xy # 4y2 # 10x ! 30 " 0
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Page 48
818 Chapter 9 Topics in Analytic Geometry
27.
(a) Parabola
(b)
"24x # 40 ± $3000x # 1600
18
y "!24x # 40" ± $!24x # 40"2 ! 4!9"!16x2 ! 30x"
2!9"
9y2 ! !24x # 40"y # !16x2 ! 30x" " 0
B2 ! 4AC " !!24"2 ! 4!16"!9" " 0 !
16x2 ! 24xy # 9y2 ! 30x ! 40y " 0
(c)
!2
!4 8
6
28. (a)
(b)
(c)
!8
!12 12
8
"4x ± $24x2 ! 48
!4
y "4x ± $16x2 ! 4!!2"!x2 ! 6"
!4
!2y2 ! 4xy # x2 ! 6 " 0
> 0 ! Hyperbola
" 16 # 8 " 24
B2 ! 4AC " !!4"2 ! 4!1"!!2" 29.
(a)
(b)
(c)
!4
!6 6
4
"8x ±$1260 ! 356x2
14
y "8x ± $!!8x"2 ! 4!7"!15x2 ! 45"
14
7y2 ! 8xy # !15x2 ! 45" " 0
" !356 ! Ellipse or circle
B2 ! 4AC " !!8"2 ! 4!15"!7"
15x2 ! 8xy # 7y2 ! 45 " 0
30. (a)
(b)
(c)
!6
!10 8
6
"!4 ! 4x" ±$!24x2 ! 92x # 416
10
y "!4 ! 4x" ± $!4x ! 4"2 ! 4!5"!2x2 # 3x ! 20"
10
5y2 # !4x ! 4"y # !2x2 # 3x ! 20" " 0
" !24 < 0 ! Ellipse or circle
B2 ! 4AC " 42 ! 4!2"!5" 31.
(a)
(b)
(c)
!6
!10 8
6
"6x ±$56x2 # 80x ! 440
!10
y "6x ± $!!6x"2 ! 4!!5"!x2 # 4x ! 22"
!10
!5y2 ! 6xy # !x2 # 4x ! 22" " 0
" 56 ! Hyperbola
B2 ! 4AC " !!6"2 ! 4!1"!!5"
x2 ! 6xy ! 5y2 # 4x ! 22 " 0
32. (a)
(c)
!7
!10 2
1
" 0 ! Parabola
B2 ! 4AC " !!60"2 ! 4!36"!25" (b)
"60x ! 9 ±$!1080x # 81
50
y "!60x ! 9" ± $!9 ! 60x"2 ! 100!36x2"
50
25y2 # !9 ! 60x"y # 36x2 " 0
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Page 49
Section 9.4 Rotation and Systems of Quadratic Equations 819
33.
(a) Parabola
(b)
(c)
!5
!4 8
3
"1 ! 4x ±$72x # 49
8
y "!1 ! 4x" ± $!4x ! 1"2 ! 4!4"!x2 ! 5x ! 3"
8
4y2 # !4x ! 1"y # !x2 ! 5x ! 3" " 0
!B2 ! 4AC " 42 ! 4!1"!4" " 0
x2 # 4xy # 4y2 ! 5x ! y ! 3 " 0 34. (a)
(b)
(c)
!2
!3 3
2
"!x ! 1 ±$!15x2 ! 14x # 65
8
y "!!x # 1" ± $!x # 1"2 ! 4!4"!x2 # x ! 4"
8
4y2 # !x # 1"y # !x2 # x ! 4" " 0
" !15 < 0 ! Ellipse or circle
B2 ! 4AC " 1 ! 4!1"!4"
35.
Two intersecting lines
y " ±4x
y2 " 16x2
!1 1 2 3
3
!2!3x
y y2 ! 16x2 " 0 36.
Point at
!4 !3 !2 !1 1 2 3 4
!4
!3
!2
2
1
3
4
x
(1, !3)
y!1, !3"
!x ! 1"2 # !y # 3"2 " 0
!x2 ! 2x # 1" # !y2 # 6y # 9" " !10 # 1 # 9
x2 # y2 ! 2x # 6y # 10 " 0
37.
Two parallel lines
x
y
!1!2!3 1 2 3
1
2
3
!2
!3
y " !x ± 1
x # y " ±1
!x # y"2 " 1
!x # y"2 ! 1 " 0
x2 # 2xy # y2 ! 1 " 0 38.
Two lines
!2 !1 1 2 3 4
!2
1
2
3
4
x
y
y " !5 ± 2$6 "x y " 5x ± 2$6x
y ! 5x " ±$24x2
!y ! 5x"2 " 24x2
y2 ! 10xy # 25x2 " 25x2 ! x2
x2 ! 10xy # y2 " 0
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Page 50
820 Chapter 9 Topics in Analytic Geometry
39.
Adding:
For
is impossible.
Solutions: !1, $3 ", !1, !$3 "x " !4
x " 1, y " ±$3.
x " 1, !4! !x # 4"!x ! 1" " 0
x2 # 3x ! 4 " 0
3x ! y2 " 0
x2 # y2 " 4 40.
No solution
"36 ± $!864
10
y "36 ± $362 ! 4!5"!108"
10
5y2 ! 36y # 108 " 0
4!27 ! y2" # 9y2 ! 36y " 0
x2 # y2 ! 27 " 0 ! x2 " 27 ! y2
4x2 # 9y2 ! 36y " 0
41.
x " !8
24x " !192
24x # 192 " 0
4x2 # y2 # 40x ! 24y # 208 " 0
!4x2 ! y2 ! 16x # 24y ! 16 " 0
43.
x " 0, 12! x!x ! 12" " 0
x2 ! 12x " 0
2x2 ! 24x " 0
x2 # y2 ! 12x ! 16y # 64 " 0
x2 ! y2 ! 12x # 16y ! 64 " 0
42.
When
Points of intersection: !6, !8", !14, !8"
y " !8
!y # 8"2 " 0
y2 # 16y # 64 " 0
!4y2 ! 64y ! 256 " 0
62 ! 4y2 ! 20!6" ! 64y ! 172 " 0
x " 6:
x " 6 or x " 14
!17x ! 238"!x ! 6" " 0
17x2 !340x # 1428 " 0
16x2 # 4y2 ! 320x # 64y # 1600 " 0 ! 16!x ! 10"2 # 4!y # 8"2 " 256
x2 ! 4y2 ! 20x ! 64y ! 172 " 0 ! !x ! 10"2 ! 4!y # 8"2 " 16
When
y " !8
!y # 8"2 " 0
y2 # 16y # 64 " 0
4y2 # 64y # 256 " 0
142 ! 4y2 ! 20!14" ! 64y ! 172 " 0
x " 14:
For
Solutions: !0, 8", !12, 8"
y " 8! !y2 # 16y ! 64 " 0
144 ! y2 ! 12!12" # 16y ! 64 " 0
x " 12:
For
Solution: !!8, 12"
! y " 12
!y ! 12"2 " 0
y2 ! 24y # 144 " 0
!y2 # 24y ! 144 " 0
!4!64" ! y2 ! 16!!8" # 24y ! 16 " 0
x " !8:
For
y " 8! !y ! 8"2 " 0
y2 ! 16y # 64 " 0
!y2 # 16y ! 64 " 0
x " 0:
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Page 51
Section 9.4 Rotation and Systems of Quadratic Equations 821
44.
When
Point of intersection: !1, 0"
x " 1
!x ! 1"2 " 0
x2 ! 2x # 1 " 0
x2 # 4!0"2 ! 2x ! 8!0" # 1 " 0
y " 0:
y " 0 or y " 3
4y!y ! 3" " 0
4y2 !12y " 0
!x2 # 2x ! 4y ! 1 " 0 ! y " ! 14!x ! 1"2
x2 # 4y2 ! 2x ! 8y # 1 " 0 ! !x ! 1"2 # 4!y ! 1"2 " 4
When
No real solution
x2 ! 2x # 13 " 0
!x2 # 2x ! 4!3" ! 1 " 0
y " 3:
45.
When
No real solution
The point of intersection is In standard form the equations are:
x2
25#
y2
16" 1
x2
4#
!y ! 12"2
64" 1
!0, 4".
16x2 " !225
16x2 # 25!!5"2 ! 400 " 0
y " !5:
y " !5 or y " 4
24!y # 5"!y ! 4" " 0
24y2 # 24y ! 480 " 0
16x2 # 25y2 ! 400 " 0
!16x2 ! y2 # 24y ! 80 " 0
When
x " 0
16x2 " 0
16x2 # 25!4"2 ! 400 " 0
y " 4:
46.
When
Points of intersection: !5, 8 # 4$21 ", !5, 8 ! 4$21 ", !!2, 8"
y " 8 ± 4$21
y2 ! 16y ! 272 " 0
y2 ! 48!5" ! 16y ! 32 " 0
x " 5:
x " 5 or x " !2
!x ! 5"!x # 2" " 0
16!x2 ! 3x ! 10" " 0
16x2 ! 48x ! 160 " 0
y2 ! 48x ! 16y ! 32 " 0 ! !y ! 8"2 ! 48x " 96
16x2 ! y2 # 16y ! 128 " 0 ! 16x2 ! !y ! 8"2 " 64
When
y " 8
!y ! 8"2 " 0
y2 ! 16y # 64 " 0
y2 ! 48!!2" ! 16y ! 32 " 0
x " !2:
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Page 52
822 Chapter 9 Topics in Analytic Geometry
47.
Two solutions: !$3, !2$3 ", !!$3, 2$3 " x2 " 3 ! x " ±$3
!2x2 # 6 " 0
2x2 ! !!2x"2 # 6 " 0
2x # y " 0 ! y " !2x
2x2 ! y2 # 6 " 0 48.
Solutions: !0, 2", !43, 23"
x " 43 ! y " 2 ! 4
3 " 23
x " 0 ! y " 2
3x!3x ! 4" " 0
9x2 ! 12x " 0
6x2 # 3!4 ! 4x # x2" ! 12 " 0
6x2 # 3!2 ! x"2 ! 12 " 0
x # y ! 2 " 0 ! y " 2 ! x
6x2 # 3y2 ! 12 " 0
49.
impossible
One solution: !8, 0"
x " 7 ! y2 " !2
x " 8 ! y2 " 0 ! !8, 0"
!x ! 8"!x ! 7" " 0
x2 ! 15x # 56 " 0
10x2 ! 150x # 560 " 0
10x2 ! 25!2x ! 16" ! 100x # 160 " 0
y2 ! 2x # 16 " 0 ! y2 " 2x ! 16
10x2 ! 25y2 ! 100x # 160 " 0 50.
From Equation 1:
In Equation 2:
Solution: !2, 3"
x " 45 ! y2 ! 6y " !321
25 No solution
! !y ! 3"2 " 0 ! y " 3
x " 2 ! y2 ! 6y " !9
!x ! 2"!5x ! 4" " 0
5x2 ! 14x # 8 " 0
!10x2 # 28x ! 16 " 0
2x2 # 4x ! 43 ! !12x2 ! 24x ! 27" " 0
2x2 # 4x ! 43 ! !3y2 ! 18y" " 0
3y2 ! 18y " 12x2 ! 24x ! 27
y2 ! 6y " 4x2 ! 8x ! 9
2x2 ! 3y2 # 4x # 18y ! 43 " 0
4x2 ! y2 ! 8x # 6y ! 9 " 0
51.
x " 0 or x " !3
x(x # 3)!x2 ! 7x # 20" " 0
x4 ! 4x3 ! x2 # 60x " 0
x4 ! 4x3 # 4x2 # 4x2 # 24x # 36 " 9x2 ! 36x # 36
x2!x2 ! 4x # 4" # 4!x2 # 6x # 9" " 9!x2 ! 4x # 4"
x2!x ! 2"2 # 4!!x ! 3"2 " 9!x ! 2"2
x2 # 4%!x ! 3x ! 2 &
2
" 9
x2 # 4y2 ! 9 " 0
xy # x ! 2y # 3 " 0 ! y "!x ! 3x ! 2
Note: has no real solution.
When
When
The points of intersection are !0, 32", !!3, 0".
y "!!!3" ! 3
!3 ! 2" 0x " !3:
y "!0 ! 30 ! 2
"32
x " 0:
x2 ! 7x # 20 " 0
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Page 53
Section 9.4 Rotation and Systems of Quadratic Equations 823
52.
When :
When :
Points of intersection: %16
!3 ! $30 ", 16
!3 # $30 "&%16
!3 # $30 ", 16
!3 ! $30 "&,
y " 1 !3 !$30
6"
3 #$306
x "3 !$30
6
y " 1 !3 #$30
6"
3 !$306
x "3 #$30
6
x "3 ±$30
6
12x2 ! 12x ! 7 " 0
5x2 ! 2x # 2x2 # 5 ! 10x # 5x2 ! 12 " 0
5x2 ! 2x # 2x2 # 5!1 ! 2x # x2" ! 12 " 0
5x2 ! 2x!1 ! x" # 5!1 ! x"2 ! 12 " 0
x # y ! 1 " 0 ! y " 1 ! x
5x2 ! 2xy # 5y2 ! 12 " 0
53. True.
If then B2 ! 4AC > 0.k < 14,
B2 ! 4AC " 1 ! 4k 54. False. See Example 2. However, A # C " A$ # C$.
55.
Asymptotes:
Intercepts: !0, 1"
x " 2, y " 0!2 !1 1 3
!4
!3
!2
2
3
4
x
(0, 1)
y
g!x" "2
2 ! x56.
Intercept:
Asymptotes:x " 2, y " !2
4 6 8
!6
!4
2
4
6
x(0, 0)
y!0, 0"
f !x" "2x
2 ! x" !2 #
42 ! x
57.
Slant asymptote:
Vertical asymptote:
Intercept: !0, 0"
!10 !5 5 10 15
!15
!10
!5
5
10
(0, 0)t
yt " 2
y " !t ! 2
h!t" "t2
2 ! t" !t ! 2 #
42 ! t
58.
Intercept:
Asymptotes:s " ±2, y " 0
%0, 12&
!4 !1 1 4
!4
!3
!2
1
2
3
4
s( )0 , 1
2
yg!s" "2
4 ! s2
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Page 54
824 Chapter 9 Topics in Analytic Geometry
59. (a)
(b)
(c) A2 " )12
!35*)
12
!35* " )!5
12!18
19*
BA " )05
6!1*)
12
!35* " )12
330
!20*
AB " )12
!35*)
05
6!1* " )!15
2597* 60. (a)
(b)
(c) A2 " )10
5!2*)
10
5!2* " )1
0!5
4*
BA " ) 3!3
28*)
10
5!2* " ) 3
!311
!31*
AB " )10
5!2*)
3!3
28* " )!12
642
!16*
61. (a)
(b)
(c) does not exist.A2
BA " ) 3!4
5*'4 !2 5( " ) 12!16
20
!68
!10
15!20
25*AB " '4 !2 5() 3
!45* " '12 # 8 # 25( " '45(
62. (a)
(b)
(c) A2 " )!2164
!219
!2
!105
20*BA " )!9
29
!149
!1
02515*
AB " ) 827
!13
!102020
26
!13* 63.
x2
4
6
8
10
!8 !6 !4 !2!2
4
y
f !x" " ,x # 3, 64.
x2
2
4
8
10
!2!2
4 6 8 10
y
f !x" " ,x ! 4, # 1
65.
!1!2!3 1 2 3
1
3
!2
!1
!3
x
y
g!x" " $4 ! x2 66.
x2
2
4
6
8
10
!2!2
4 6 8 10
y
g!x" " $3x ! 2 67.
!2!4!6!8!10 2 6 8 10
2468
10121416
!4
x
y
h!t" " !!t ! 2"3 # 3
68.
x
y
!1!2!3!4!6 1 2
!4
!3
!2
!1
1
2
3
4
h!t" " 12!t # 4"3 69.
!1!2!3 1 2 5 6 7
123
!2!3!4
!7
x
y
f !t" " -t ! 5. # 1 70.
!1!2!3!4 1
1
2
3
4
2 3 4
!3
!2
x
y
f !t" " !2-t. # 3
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Page 55
Section 9.5 Parametric Equations 825
71.
! 45.11
! 12"8#"12# sin 110"
Area ! 12ab sin C 72.
! 187.94
! 12"25#"16# sin 70"
Area ! 12ac sin B
73.
! 48.60
! $392 "17
2 #"32#"19
2 #Area ! $s"s # a#"s # b#"s # c#
s ! 12"11 $ 18 $ 10# ! 39
2 74.
! 310.39
! $852 "39
2 #"152 #"31
2 #Area ! $s"s # a#"s # b#"s # c#
s ! 12"23 $ 35 $ 27# ! 85
2
Section 9.5 Parametric Equations
! If f and g are continuous functions of t on an interval I, then the set of ordered pairs is a planecurve C. The equations and are parametric equations for C and t is the parameter.
! You should be able to graph plane curves with your graphing utility.
! To eliminate the parameter:
Solve for t in one equation and substitute into the second equation.
! You should be able to find the parametric equations for a graph.
y ! g"t#x ! f"t#"f"t#, g"t##
Vocabulary Check
1. plane curve, parametric equations, parameter 2. orientation
3. eliminating, parameter
1.
line
Matches (c).
y ! x $ 2,
y ! t $ 2
x ! t 2.
Parabola opening to the rightMatches (d).
x ! "y $ 2#2
y ! t # 2 ! t ! y $ 2
x ! t2 3.
parabola,
Matches (b).
x " 0y ! x2,
y ! t
x ! $t
4.
Matches (a).
y ! t $ 2 ! y !1x
$ 2
x !1t ! t !
1x
5.
Matches (f).
y !12
ex # 2
y !12
t # 2
x ! ln t # t ! ex 6.
Exponential curve on
Matches (e).
x $ 0
y ! ex2%4
y ! et
x ! #2$t ! t ! & x#2'
2
!x2
4
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Page 56
826 Chapter 9 Topics in Analytic Geometry
11.
!1
!2
!1
!2 1 2
2
x
y
y ! #4x
x ! t, y ! #4t 12.
or
2
4
2
4
6
6!6
!6
!4
!4!2
x
y
x # 2y ! 0y !12
x
x ! t, y !12
t
7.
(a)
(b) Graph by hand.
Note: x " 0
!1
!2
!1
!2 1 2
1
2
x
y
x ! $t, y ! 2 # t
(c)
(d)
Parabola
In part (c), x " 0.
!1
!2
!1
!2 1 2
1
x
yy ! 2 # t ! 2 # x2,
!3
3
5!4
0 1 2 3 4
0 1 2
2 1 0 #2#1y
$3$2x
t
8.
(a)
(b)
!1!2!3!4!5 1 2 3 5
12345
!2!3!4!5
x
y
x ! 4 cos2 %, y ! 4 sin %
(c)
(d)
parabola
The graph is an entireparabola rather thanjust the right portion.
!1!2!3!4!5 1 2 3 5
123
5
!2!3
!5
x
y x4
$ y2
16! 1,
4x $ y2 ! 16 cos2 % $ 16 sin2 % ! 16
10!8
!6
6
0
0 2 4 2 0
0 42$2#2$2#4y
x
&2
&4
#&4
#&2
%
9. The graph opens upward, contains and is oriented left to right. Matches (b).
"1, 0#, 10. The orientation of the graph is clockwise and thecenter is Matches (c)."2, 3#.
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Page 57
Section 9.5 Parametric Equations 827
13.
!1!2!3!4!6 1 2!1
!2
!3
1
2
4
5
x
y
y !23
x $ 3
y ! 2&x $ 33 ' $ 1
t !x $ 3
3
x ! 3t # 3, y ! 2t $ 1 14.
2
4
2
4
6
6 8
!4
!4!2
!2x
y
3x $ 2y # 13 ! 0
y ! 2 $ 3&3 # x2 '
x ! 3 # 2t, y ! 2 $ 3t 15.
1 2
5
3!3 !2 !1!1
x
y
y ! 16x2
y ! "4x#2
x !14
t, y ! t2
16.
2 4
2
4
6
6!6
!6
!4 !2x
y
y ! x3
x ! t, y ! t3 17.
!1!2 1 2 3 4 5 6!1
1
2
4
3
x
y
y ! "x # 2#2
t ! x # 2
x ! t $ 2, y ! t2 18.
2 4
2
6 8 10!2!2
!4
!6
!8
!10
x
y
y ! 1 # x2, x " 0
y ! 1 # t
x ! $t
20.
Eliminating the parameter and
2
6
4
2
8
10
6 8 10!2!2
y
x
! (y # 3(. ! (" y # 2# # 1(
x ! (t # 1(t ! y # 2t,
y ! t $ 2
x ! (t # 1(19.
2 4
6
4
8
10
6 8 10!2!2
x
y
!12(x # 4(
! (x2 # 2( t !
x2
! y ! (t # 2(
x ! 2t, y ! (t # 2(
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Page 58
828 Chapter 9 Topics in Analytic Geometry
21.
ellipsex2
4$
y2
9! 1,
x2
4$
y2
9! cos2 % $ sin2 % ! 1
&x2'
2! cos2 %, &y
3'2
! sin2 %
!1!3!4 1
1
2
4
3 4
!2
!4
!1
x
y
x ! 2 cos %, y ! 3 sin % 22.
x2 $y2
16! 1, ellipse
x2 $ &y4'
2! cos2 % $ sin2 % ! 1
!2!3!4!5 2 3 4 5
45
!4!5
x
y
x ! cos %, y ! 4 sin %
23.
!1!2 321 4 5 6 7 8
123456789
10
x
y
y !1x3 , x > 0, y > 0
y ! &1x'
3
y ! e3t ! y ! "et#3
x ! e#t ! 1x
! et 24.
1
1 2 3 4 5
2
3
4
5
x
y
y2 ! x, y > 0; y ! $x, x > 0
y ! et ! y2 ! e2t
x ! e2t
25.
!1
!3!2
!4!5
!2 32 4 5 6 7 8
12345
x
y
y ! ln x
y ! ln"x1%3#3
y ! 3 ln t ! y ! ln t3
x ! t3 ! x1%3 ! t 26.
1
1!1!2 2!3 3
2
3
4
5
6
x
y
y ! 2t2 ! 2"12 e x#2 ! 1
2 e2x
x ! ln 2t ! ex ! 2t ! t ! 12ex
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Page 59
Section 9.5 Parametric Equations 829
27.
8!1
!5
1
x ! 4 $ 3 cos %, y ! #2 $ sin % 28.
!6
10!2
2
x ! 4 $ 3 cos %, y ! #2 $ 2 sin %
29.
12!12
!8
8
x ! 4 sec %, y ! 2 tan % 30.
y ! tan %
!4
4
6!6
x ! sec %
31.
y ! ln"t2 $ 1#
!8
8
12!12
x !t2
32.
y ! 0.4t2
!5
35
20!40
x ! 10 # 0.01et
33. By eliminating the parameters in (a)–(d), we get . They differ from each other in restricted domain and in orientation.
(a) Domain:
Orientation: Left to right
(c) Domain:
Orientation: Right to left
0 < x < '
#' < x < '
y ! 2x $ 1
(b) Domain:
Orientation: Depends on
(d) Domain:
Orientation: Left to right
0 < x < '
%
#1 $ x $ 1
34. Each curve represents a portion of the line
(a)
Orientation: Left to right
(c)
Orientation: Left to right
y ! 3 # t ! 3 # &x # 22 ' ! 4 #
x2
x ! 2"t $ 1#, #' < x < '
y ! 4 # $t ! 4 #x2
, y $ 4
x ! 2$t, x " 0
2y $ x # 8 ! 0.
(b)
Orientation: Left to right
(d)
Orientation: Left to right for
Right to left for t > 0
t $ 0
y ! 4 $ t2 ! 4 #x2
x ! #2t2, x $ 0
y ! 4 # 3$t ! 4 #x2
x ! 2 3$t, #' < x < '
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Page 60
830 Chapter 9 Topics in Analytic Geometry
49.
!4
4
6!6
x ! 2 cot %, y ! 2 sin2 % 50.
!4
4
6!6
y !3t2
1 $ t3x !3t
1 $ t3,
35.
! y # y1 ! &y2 # y1
x2 # x1'"x # x1#
y ! y1 $ & x # x1
x2 # x1'"y2 # y1#
t !"x # x1#"x2 # x1#
37.
"x # h#2
a2 $"y # k#2
b2 ! 1
x # ha
! cos %, y # k
b! sin %
y ! k $ b sin %
x ! h $ a cos %
36.
"x # h#2 $ " y # k#2 ! r2
cos2 % $ sin2 % ! "x # h#2
r2 $" y # k#2
r2 ! 1
y # kr
! sin %"x # h#
r! cos %,
y ! k $ r sin %
x ! h $ r cos %
38.
sec2 % # tan2 % !"x # h#2
a2 #"y # k#2
b2 ! 1
y # kb
! tan %x # h
a! sec %,
y ! k $ b tan %
x ! h $ a sec %
39.
y ! y1 $ t"y2 # y1# ! 4 $ t"#3 # 4# ! 4 # 7t
x ! x1 $ t"x2 # x1# ! 1 $ t"6 # 1# ! 1 $ 5t 40.
y ! k $ r sin % ! 5 $ 4 sin %
x ! h $ r cos % ! 2 $ 4 cos %
41. and
The center is so and
so and
This solution is not unique.y ! 3 sin %.
x ! 5 cos %cos2 % $ sin2 % ! 1 !x2
52 $y2
32,
k ! 0.h ! 0"0, 0#,
b ! $a2 # c2 ! 3.a ! 5, c ! 4, 42. and
The center is so and
so and
x ! $3 tan %.
y ! sec %sec2 % # tan2 % ! 1 !y2
1#
x2
3,
k ! 0.h ! 0"0, 0#,
b ! $c2 # a2 ! $3.a ! 1, c ! 2,
43.
Answers will vary.
x !15
t, y ! t # 3
x ! t, y ! 5t # 3
y ! 5x # 3 44.
Answers will vary.
x ! 2t, y ! 4 # 14t
x ! t, y ! 4 # 7t
y ! 4 # 7x 45.
Sample answers:
x ! t3, y !1t3
x ! t, y !1t
y !1x
48.
Sample answers:
x !12
t, y !t3
8$ t
x ! t, y ! t3 $ 2t
y ! x3 $ 2x46.
Sample answers:
x ! 2t, y !14t
x ! t, y !12t
y !1
2x47.
Sample answers:
x ! 2t, y ! 24t2 # 5
x ! t, y ! 6t2 # 5
y ! 6x2 # 5©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
Page 61
Section 9.5 Parametric Equations 831
51. Matches (b). 52. Matches (c). 53. Matches (d). 54. Matches (a).
55.
(a)
(b)
It is not a home run because when x ! 400.y < 10
0
30
4500
! 3 $ 38.0t # 16t2
y ! 3 $ "146.67 sin 15"#t # 16t2
x ! "146.67 cos 15"#t ! 141.7t
% ! 15"
y ! 3 $ "146.67 sin %#t # 16t2
x ! "146.67 cos %#t
! 146.67 ft%sec
100 miles%hour !100 mi(hr ) 5280 ft(mi
3600 sec(hr
x ! "v0 cos %#t, y ! h $ "v0 sin %#t # 16t2
56. (a)
(c) The horizontal distance is approximately 342.25 feet.
(d) You could use the Quadratic Formula to findthe zeros of The larger zero, 4.255, gives feet.x ! 342.25
y ! #16t2 $ "105 sin 40"#t $ 2.5.
! 2.5 $ "105 sin 40"#t # 16t2
y ! h $ "v0 sin %#t # 16t2
x ! "v0 cos %#t ! "105 cos 40"#t
(c)
Yes, it is a home run because when
(d) is the minimum angle.% ! 19.4"
x ! 400.y > 10
0
60
5000
! 3 $ 57.3t # 16t2
y ! 3 $ "146.67 sin 23"#t # 16t2
x ! "146.67 cos 23"#t ! 135.0t
% ! 23"
(b)
The maximum height is approximately 73.68 feet, when seconds.t ! 2.109
400
80
00
57. (a)
y ! 7 $ "sin 35"#v0t # 16t2
x ! "cos 35"#v0 t (b) If the ball is caught at time then:
! v0 !90
t1 cos 35"! 54.09 ft(sec
! t1 ! 2.03 seconds
! 16t12 ! 90 tan 35" $ 3
v0t1 !90
cos 35" ! #3 ! "sin 35"# 90
cos 35"# 16t12
4 ! 7 $ "sin 35"#v0t1 # 16t12.
90 ! "cos 35"#v0t1
t1,
(c)
Maximum height
(d) From part (b), t1 ! 2.03 seconds.
! 22 feet
0
24
900
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Page 62
832 Chapter 9 Topics in Analytic Geometry
58. (a)
(b)
The maximum height is approximately 66.25 feet when seconds.
(c) The horizontal distance is approximately 222.35 feet.
(d) You could solve the equation for Then, feet.x ! 222.35
t ! 4.0696.y ! "85 sin 50"#t # 16t2 ! 0
t ! 2.035
240
80
00
y ! h $ "v0 sin %#t # 16t2 ! "85 sin 50"#t # 16t2
x ! "v0 cos %#t ! "85 cos 50"#t
59. True
first set
second set
y ! 9t2 $ 1 ! "3t#2 $ 1 ! x2 $ 1
x ! 3t
y ! t2 $ 1 ! x2 $ 1
x ! t
60. False. The graph of represents theportion of the line in the first quadrant.y ! x
x ! t2, y ! t2
61. False. For example, and does not represent as a function of x.y
y ! tx ! t2 62. False. The equations represent a line.
63. Sample answer:
y ! #2 sin %
x ! cos % 64. The graph is the same, but the orientation isreversed.
65.
Symmetric about the -axis
Even function
y
f"#x# !4"#x#2
"#x#2 $ 1!
4x2
x2 $ 1! f"x#
68.
No symmetry
Neither even nor odd
y ! x2 # 4x
"x # 2#2 ! y $ 4
66.
No symmetry
Neither even nor odd
f "x# ! $x, x " 0
67.
No symmetry
Neither even nor odd
y ! ex * e#x; e#x * #ex
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Page 63
Section 9.6 Polar Coordinates
Section 9.6 Polar Coordinates 833
! In polar coordinates you do not have unique representation of points. The point can be represented by or by where n is any integer. The pole is represented by where
is any angle.
! To convert from polar coordinates to rectangular coordinates, use the following relationships.
! To convert from rectangular coordinates to polar coordinates, use the following relationships.
If is in the same quadrant as the point , then r is positive. If is in the opposite quadrant as the pointthen r is negative.
! You should be able to convert rectangular equations to polar form and vice versa.
!x, y", !!x, y"!
tan ! " y#x
r " ±$x2 # y2
y " r sin !
x " r cos !
!!0, !"!$r, ! ± !2n # 1"%"!r, ! ± 2n%"
!r, !"
1. Polar coordinates:
Rectangular coordinates: !0, 4"
y " 4 sin%%
2& " 4
x " 4 cos%%
2& " 0
%4, %
2& 2. Polar coordinates:
Rectangular coordinates: !0, $4"
x " 4 cos%3%
2 & " 0, y " 4 sin%3%
2 & " $4
%4, 3%
2 &
3. Polar coordinates:
Rectangular coordinates: %$22
, $22 &
y " $1 sin%5%
4 & "$22
x " $1 cos%5%
4 & "$22
%$1, 5%
4 & 4. Polar coordinates:
Rectangular coordinates: !$2, $$2 "
y " 2 sin%$%4& " 2%$
$22 & " $$2
x " 2 cos%$%4& " 2%$2
2 & " $2
%2, $%4&
Vocabulary Check1. pole 2. directed distance, directed angle 3. polar
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Page 64
834 Chapter 9 Topics in Analytic Geometry
5.
Three additional representations:
%$3, 5%6
$ %& " %$3, $%6&
%$3, 5%6
# %& " %$3, 11%
6 &%3,
5%6
$ 2%& " %3, $7%6 &
01 2 3
!2
3, 6!5( (
6.
Three additional points:
%2, $5%
4 &, %$2, 7%4 &, %$2,
$%4 &
01 2 3
!2
2,43!( (
7.
Three additional representations:
%1, $%3
$ %& " %1, $4%3 &
%1, $%3
# %& " %1, 2%3 &
%$1, $%3
# 2%& " %$1, 5%3 &
01 2 3
"1, "3!( (
!2 8.
Three additional points:
%3, $%6&%$3,
5%6 &, %3,
11%6 &,
01 2 3 4
"3, " )) 76!
2!
9.
Three additional representations:
%$$3, 11%
6 &%$$3, $%6&,%$3, $
7%6 &,
0
3, )) 56!
1 2 3
2! 10.
Three additional points:
%$5$2, 7%6 &%$5$2, $
5%6 &,%5$2,
%6&,
0
( )"2,5
1 3 5 7 9
116!2
!
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Page 65
Section 9.6 Polar Coordinates 835
11.
Three additional representations:
%32
, %2&, %$
32
, 3%2 &, %$
32
, $%2&
32
0
, "
1 2 3
)) 32!2
! 12.
is the origin.
Three additional points:
Any angle will do since r " 0."!
%0, 3%4 &, %0,
$5%4 &, %0,
7%4 &
%0, $%
4 &
01 2 3
!2
0,4!( ("
13. Polar coordinates:
Rectangular coordinates:
2 4 60
2!
!2, $2$3"
y " 4 sin%$%
3& " $2$3
x " 4 cos%$%
3& " 2
%4, $%
3& 14. Polar coordinates:
Rectangular coordinates:
1 2 3
2!
0
!$$3, $1"
y " 2 sin 7%6
" 2%$12& " $1
x " 2 cos 7%6
" 2%$$32 & " $$3
%2, 7%6 &
15. Polar coordinates:
Rectangular coordinates:
1 2 30
2!
%$22
, $22 &
y " $1 sin%$3%
4 & "$22
x " $1 cos%$3%
4 & "$22
%$1, $3%
4 & 16. Polar coordinates:
Rectangular coordinates:
01 2 3
2!
%32
, 3$3
2 &
y " $3 sin%$2%3 & " $3%$
$32 & "
3$32
x " $3 cos%$2%3 & " $3%$
12& "
32
%$3, $2%3 & " %3,
%3&
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Page 66
836 Chapter 9 Topics in Analytic Geometry
17. Polar coordinates: (origin!)
Rectangular coordinates:
1 2 30
2!
!0, 0"
y " 0 sin%$7%
6 & " 0
x " 0 cos%$7%
6 & " 0
%0, $7%
6 & 18. Polar coordinates: (origin!)
Rectangular coordinates:
01 2 3
2!
!0, 0"
y " 0 sin 5%4
" 0
x " 0 cos 5%4
" 0
%0, 5%4 &
20. Polar coordinates:
Rectangular coordinates:
1 2 30
2!
!$0.0024, 3"
y " $3 sin!$1.57" ' 3.000
x " $3 cos!$1.57" ' $0.0024
!$3, $1.57"19. Polar coordinates:
Rectangular coordinates:
1 2 30
2!
!$1.004, 0.996"
y " $2 sin!2.36" ' 0.996
x " $2 cos!2.36" ' $1.004
!$2, 2.36"
21. !r, !" " %2, 2%
9 & ! !x, y" " !1.53, 1.29" 22. !r, !" " %4, 11%
9 & ! !x, y" " !$3.06, $2.57"
23. !r, !" " !$4.5, 1.3" ! !x, y" " !$1.204, $4.336" 24. !r, !" " !8.25, 3.5" ! !x, y" " !$7.726, $2.894"
25. !r, !" " !2.5, 1.58" ! !x, y" " !$0.02, 2.50" 26. !r, !" " !5.4, 2.85" ! !x, y" " !$5.17, 1.55"
27. !r, !" " !$4.1, $0.5" ! !x, y" " !$3.60, 1.97" 28. !r, !" " !8.2, $3.2" ! !x, y" " !$8.19, 0.48"
29. Rectangular coordinates:
Polar coordinates: !7, %", !$7, 0"
r " 7, tan ! " 0, ! " 0
"1"2"4"6"8 "3"5"7"9
1
1
"2"3"4"5
2
3
45
x
y!$7, 0"
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Page 67
Section 9.6 Polar Coordinates 837
30. Rectangular coordinates:
Polar coordinates:
1 2 3"3
"3
" 4
"5
"6
"2
"2"1
"1x
y
%5, 3%
2 &, %$5, %
2&
r " 5, tan ! undefined, ! "%
2
!0, $5" 31. Rectangular coordinates:
Polar coordinates:
1
2
1
2
3
3"3
"3
"2
"2"1
"1x
y
%$2, %
4&, %$$2, 5%
4 &
r " $2, tan ! " 1, ! "%
4
!1, 1"
32. Rectangular coordinates:
Polar coordinates:
2
1
1 2"3"4
"4
"3
"2
"2"1
"1x
y
%3$2, 5%
4 &, %$3$2, %
4&
r " 3$2, tan ! " 1, ! "%
4
!$3, $3"
34. Rectangular coordinates:
Polar coordinates:
x
y
"1"2 1 2 3 4 5 6"1
"2
"3
"4
"5
"6
1
2
%2, 11%
6 &, %$2, 5%
6 &
! "11%
6tan ! "
$1$3
,
r " $3 # 1 " 2
!$3, $1"
33. Rectangular coordinates:
Polar coordinates:
1
2
1
2
3
3"3
"3
"2
"2"1
"1x
y
%$6, 5%
4 &, %$$6, %
4&
r " $3 # 3 " $6, tan ! " 1, ! "%
4
!$$3, $$3"
35.
Polar coordinates:
"3 3 6 9 12
"3
3
6
9
12
x
y
!10.8, 0.983", !$10.8, 4.124"
tan ! "96
"32
! ! ' 0.983
r " $62 # 92 " $117 ' 10.8
!x, y" " !6, 9"
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Page 68
838 Chapter 9 Topics in Analytic Geometry
36. Rectangular coordinates:
Polar coordinates:
4
2
"2"4"6
6
8
10
12
4 6 82"2
x
y
!$13, 4.318" !13, 1.176",
! ' 1.176tan ! " 125 ,r " $25 # 144 " 13,
!5, 12" 37.
!r, !" ' !$13, $0.588" ! " arctan!$2
3" ' $0.588
!x, y" " !3, $2" ! r " $32 # !$2"2 " $13
38. !x, y" " !$5, 2" ! !r, !" " !5.39, 2.76"
42. !x, y" " %74
, 32& ! !r, !" " !2.30, 0.71"
39.
!r, !" ' !$7, 0.857"
! " arctan% 2$3& ' 0.857
!x, y" " !$3, 2" ! r " $3 # 22 " $7
40. !x, y" " !3$2, 3$2" ! !r, !" " %6, %
4& ' !6.0, 0.785"
41.
!r, !" ' %176
, 0.490&
! " arctan%4#35#2& ' 0.490
!x, y" " %52
, 43& ! r "$%5
2&2
# %43&
2
"176
43.
r " 3
r2 " 9
x2 # y2 " 9 44.
r " 4
r2 " 16
x2 # y2 " 16 45.
r " 4 csc !
r sin ! " 4
y " 4
47.
r " 8 sec !
r cos ! " 8
x " 846.
! "%4
tan ! " 1
sin ! " cos !
r sin ! " r cos !
y " x 48.
r " a sec !
r cos ! " a
x " a
49.
r "2
6 sin ! $ 3 cos !
r!3 cos ! $ 6 sin !" " $2
3r cos ! $ 6r sin ! " $2
3x $ 6y # 2 " 0 50.
r "2
4 cos ! # 7 sin !
r!4 cos ! # 7 sin !" " 2
4r cos ! # 7r sin ! $ 2 " 0
4x # 7y $ 2 " 0
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Page 69
Section 9.6 Polar Coordinates 839
51.
r2 " 8 csc 2!
r2 sin 2! " 8
r2!2 cos ! sin !" " 8
r2 cos ! sin ! " 4
!r cos !"!r sin !" " 4
xy " 4 52.
r2 " 12 sec ! csc ! " csc!2!"
2r2 " sec ! csc !
2r cos ! & r sin ! " 1
2xy " 1
53.
r2 " 9 cos!2!"
r2 " 9!cos2 ! $ sin2 !"
!r2"2 " 9!r2 cos2 ! $ r2 sin2 !"
!x2 # y2"2 " 9!x2 $ y2" 54.
or r "$4
1 # cos !r "
41 $ cos !
r " ±!r cos ! # 4"
!r cos ! # 4"2 " r2
r2 cos2 ! # 8r cos ! # 16 " r2
r2!1 $ cos2 !" $ 8r cos ! $ 16 " 0
r2 sin2 ! $ 8r cos ! $ 16 " 0
y2 $ 8x $ 16 " 0
55.
r " 6 cos !
r2 " 6r cos !
r2 $ 6r cos ! " 0
x2 # y2 $ 6x " 0 56.
r " 8 sin !
r!r $ 8 sin !" " 0
r2 $ 8r sin ! " 0
x2 # y2 $ 8y " 0 57.
r " 2a cos !
r!r $ 2a cos !" " 0
r2 $ 2ar cos ! " 0
x2 # y2 $ 2ax " 0
58.
r " 2a sin !
r!r $ 2a sin !" " 0
r2 $ 2a r sin ! " 0
x2 # y2 $ 2ay " 0 59.
" tan2 ! sec !
r "sin2 !cos3 !
sin2 ! " r cos3 !
!r sin !"2 " !r cos !"3
y2 " x3 60.
r "cos2 !sin3 !
" cot2 ! csc !
r2 cos2 ! " r3 sin3 !
x2 " y3
61.
x2 # y2 $ 6y " 0
x2 # y2 " 6y
r2 " 6r sin !
r " 6 sin ! 62.
x2 # y2 " 2x
r2 " 2r cos !
r " 2 cos ! 63.
y " $3x
$3 "yx
tan ! " tan 4%3
"yx
! "4%3
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Page 70
840 Chapter 9 Topics in Analytic Geometry
64.
y # $3x " 0
yx
" $$3
tan ! " tan 5%3
" $$3
! "5%3
65.
y "$$3
3x
$$3
3"
yx
tan ! " tan 5%6
"yx
! "5%6
66.
y "$$3
3x
$$3
3"
yx
tan ! " tan 11%
6"
yx
! "11%
6
67. vertical line
x " 0
! "%2
, 68. horizontal line
y " 0
! " %, 69.
x2 # y2 " 16
r2 " 16
r " 4
70.
x2 # y2 " 100
r2 " 100
r " 10 71.
y " $3
r sin ! " $3
r " $3 csc ! 72.
x " 2
r cos ! " 2
r " 2 sec !
73.
!x2 # y2"3 " x2
x2 # y2 " x2#3
!x2 # y2"3#2 " x
r3 " r cos !
r2 " cos ! 74.
!x2 # y2)2 " 2xy
r4 " 2xy
r2 " 2 %yr&%
xr& "
2xyr2
r2 " sin 2! " 2 sin ! cos !
75.
!x2 # y2"2 " 6x2y $ 2y3
!x2 # y2"2 " 6!x2 # y2"y $ 8y3
r4 " 6r3 sin ! $ 8r3 sin3 !
r " 2!3 sin ! $ 4 sin3 !"
r " 2 sin 3! 76.
!x2 # y2"3#2 " 3!x2 $ y2" or !x2 # y2"3 " 9!x2 $ y2"2
r3 " 3!r2 cos2 ! $ r2 sin2 !"
r " 3!cos2 ! $ sin2 !"
r " 3 cos 2!
77.
y2 " 2x # 1
x2 # y2 " 1 # 2x # x2
$x2 # y2 $ x " 1
r $ r cos ! " 1
r "1
1 $ cos !78.
x2 # 4y $ 4 " 0
x2 # y2 " 4 $ 4y # y2
x2 # y2 " !2 $ y"2
$x2 # y2 # y " 2
r # r sin ! " 2
r "2
1 # sin !©
Hou
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Com
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. All
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serv
ed.
Page 71
Section 9.6 Polar Coordinates 841
79.
4x2 $ 5y2 $ 36y $ 36 " 0
4x2 # 4y2 " 36 # 36y # 9y2
4!x2 # y2" " !6 # 3y"2
2!±$x2 # y2" " 6 # 3y
2r " 6 # 3r sin !
r!2 $ 3 sin !" " 6
r "6
2 $ 3 sin !80.
2x $ 3y " 6
1 "6
2x $ 3y
r "6r
2x $ 3y
r "6
2!x#r" $ 3!y#r"
r "6
2 cos ! $ 3 sin !
81.
The graph is a circle centered at the origin with radius 7.
x2 # y2 " 49
r2 " 49
x
y
"2"4"8 2 4 6 8"2
"4
"6
"8
2
4
6
8
r " 7 82.
Circle of radius 8 centered at origin
x2 # y2 " 64
r2 " 64
2
42
4
6
10
6 10"6"10 "2
"6
"10
"4
"4x
y r " 8
83.
The graph is the linewhich makes an
angle of withthe positive axis.x-
! " %#4y " x,
y " x
tan ! " tan %4
" 1 "yx
1
2
1
2
3
3"3
"3
"2
"2"1
"1x
y ! "%4
84.
Line through origin making angle of with positive x-axis
%#6
3y $ $3x " 0
yx
" tan ! " tan 7%
6"
$33
x
y
"2"3 1 2 3"1
"2
"3
1
2
3
! "7%
6
85.
Vertical line
x $ 3 " 0
x " 3
r cos ! " 32
1
2 41
3
"3
"2
"2"1
"1x
yr " 3 sec ! 86.
Horizontal line through!0, 2"
y $ 2 " 0
y " 2
r sin ! " 2
1
21
3
4
3"3
"2
"2"1
"1x
y r " 2 csc !
87. True, the distances from the origin are the same. 88. False. For instance when any value of gives the same point.
!r " 0,
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Page 72
842 Chapter 9 Topics in Analytic Geometry
89. (a) where and
where and
Then and Thus,
(b) If the points are on the same line through the origin. In this case,
(c) If the Pythagorean Theorem.
(d) For instance, gives gives (Same!)d ' 2.053.%$3, 7%
6 &, %$4, 4%
3 &d ' 2.053 and%3, %
6&, %4, %
3&!1 $ !2 " 90', d " $r1
2 # r22,
d " $r12 # r2
2 $ 2r1r2 cos!0" " $!r1 $ r2"2 " (r1 $ r2(.!1 " !2,
" $r12 # r2
2 $ 2r1r2 cos!!1 $ !2".
" $r12 # r2
2 $ 2!r1r2 cos !1 cos !2 # r1r2 sin !1 sin !2"
" $!x12 # y1
2" # !x22 # y2
2" $ 2!x1x2 # y1y2"
" $x12 $ 2x1x2 # x2
2 # y12 $ 2y1y2 # y2
2
d " $!x1 $ x2"2 # !y1 $ y2"2
x22 # y2
2 " r22.x1
2 # y12 " r1
2 cos2 !1 # r12 sin2 !1 " r1
2
y2 " r2 sin !2.x2 " r2 cos !2!r2, !2" " !x2, y2"
y1 " r1 sin !1.x1 " r1 cos !1!r1, !1" " !x1, y1"
90. Answers will vary.
91.
C ' 180' $ 30.7' $ 48.2' ' 101.1'
B ' 48.2'
cos B "a2 # c2 $ b2
2ac"
132 # 252 $ 192
2!13"!25" " 0.66615
A ' 30.7'
cos A "b2 # c2 $ a2
2bc"
192 # 252 $ 132
2!19"!25" " 0.86 92.
c "a sin Csin A
' 15.17
C " 180' $ A $ B ' 141.9'
sin B "b sin A
a' 0.2440 ! B ' 14.1'
A " 24', a " 10, b " 6
93.
bsin B
"c
sin C ! b "
c sin Bsin C
"12 sin!86'"
sin!38'" ' 19.44
asin A
"c
sin C ! a "
c sin Asin C
"12 sin!56'"
sin!38'" ' 16.16
B " 180' $ 56' $ 38' " 86' 94.
A " 180' $ B $ C ' 41.9'
sin C " c sin B
b' 0.9214 ! C ' 67.1'
b2 " a2 # c2 $ 2ac cos B ' 885.458 ! b ' 29.76
B " 71', a " 21, c " 29
95.
A " 180' $ B $ C " 119.1'
! B ' 25.9'
b
sin B"
csin C
! sin B "b sin C
c"
4 sin!35'"5.25
c ' 5.25
' 27.57
" 82 # 42 $ 2!8"!4" cos!35'"
c2 " a2 # b2 $ 2ab cos C 96.
a "b sin Asin B
' 53.06
A " 180' $ B $ C ' 66.5'
sin C "c sin B
b' 0.7605 ! C ' 49.5'
B " 64', b " 52, c " 44©
Hou
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ifflin
Com
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. All
right
s re
serv
ed.
Page 73
Section 9.6 Polar Coordinates 843
97. By Cramer’s Rule,
,
Solution: !2, 3"
y "Dy
D"
$48$16
" 3x "Dx
D"
$32$16
" 2
Dy " ( 5$3
$11$3( " $15 $ 33 " $48
Dx " ($11$3
$71( " $11 $ 21 " $32
D " ( 5$3
$71( " 5 $ 21 " $16
98. By Cramer’s Rule,
,
Solution: %$526
, 5526&
y "Dy
D"
$55$26
"5526
x "Dx
D" $
526
,
Dy " (34 10$5( " $15 $ 40 " $55
Dx " ($10$5
5$2( " $20 $ !$25" " 5
D " (34 5$2( " $6 $ 20 " $26
99. By Cramer’s Rule,
, ,
Solution: !0, 0, 0"
c "Dc
D" 0b "
Db
D" 0a "
Da
D" 0
Dc " (321 $21
$3
000( " 0
Db " (321 000
1$3
9( " 0
Da " (000 $21
$3
1$3
9( " 0
D " (321 $21
$3
1$3
9( " 35
100. By Cramer’s Rule,
Solution: %29589
, 84489
, $67289 &
w "Dw
D"
672$89
"$672
89
v "Dv
D"
$844$89
"84489
,u "Du
D"
$295$89
"29589
,
Dw " (518 7$2$2
1570( " 672
Dv " (518 1570
9$3
1( " $844
Du " (1570
7$2$2
9$3
1( " $295
D " (518 7$2$2
9$3
1( " $89
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Page 74
844 Chapter 9 Topics in Analytic Geometry
101. By Cramer’s Rule,
Solution: !2, $3, 3 "
z "Dz
D"
$45$15
" 3
y "Dy
D"
45$15
" $3,x "Dx
D"
$30$15
" 2,
Dz " ($125
134
1$2
4( " $45
Dy " ($125
1$2
4
212( " 45
Dx " ( 1$2
4
134
212( " $30
D " ($125
134
212( " $15
102. Cramer’s Rule does not apply because
Use elimination to solve the system.
Let then and
Solution: !$2a # 32, 2a # 1, a"
x1 " $2a # 32.x2 " 2a # 1x3 " a,
)200
010
4$2
0
!!!
310*
$R2 # R1
2R2 # R3
)200
11
$2
2$2
4
!!!
41
$2*"$R1 # R2
$R1 # R3
)222
12
$1
206
!!!
452*"
D " (222 12
$1
206( " 0.
103. Points:
The points are not collinear.
( 46
$2
$3$7$1
111( " $20 ( 0
!4, $3", !6, $7", !$2, $1" 104. Points:
($204
41
$5
111( " 0 ! collinear
!$2, 4", !0, 1", !4, $5"
105. Points:
The points are collinear.
($6$11.5
$4$3
$2.5
111( " 0
!$6, $4", !$1, $3", !1.5, $2.5" 106. Points:
($2.3$0.5
1.5
50
$3
111( " 4.6 ! not collinear
!$2.3, 5", !$0.5, 0", !1.5, $3"
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Page 75
Section 9.7 Graphs of Polar Equations 845
3. Lemniscate
Section 9.7 Graphs of Polar Equations
! When graphing polar equations:
1. Test for symmetry
(a) Replace by
(b) Polar axis: Replace by
(c) Pole: Replace by
(d) is symmetric with respect to the line
(e) is symmetric with respect to the polar axis.
2. Find the ! values for which is maximum.
3. Find the ! values for which
4. Know the different types of polar graphs.
(a) Limaçons (b) Rose curves, (c) Circles (d) Lemniscates
! You should be able to graph polar equations of the form with your graphing utility. If your utilitydoes not have a polar mode, use
in parametric mode.
y " f !t" sin t
x " f !t" cos t
r " f !!"r " a
r2 " a2 sin 2!r " a sin !r " a sin n!r " a ± b sin !
r2 " a2 cos 2!r " a cos !r " a cos n!r " a ± b cos !
n ! 2
r " 0.#r#
r " f !cos !"! " #$2.r " f !sin !"
!r, # $ !" or !%r, !".!r, !"!r, %!" or !%r, # % !".!r, !"
!r, # % !" or !%r, %!".!r, !"! " #$2:
Vocabulary Check
1. 2. polar axis 3. convex limaçon
4. circle 5. lemniscate 6. cardioid
! "#2
1. is a rose curve.r " 3 cos 2! 2. Cardioid
5. is a rose curve.r " 6 sin 2! 6. Limaçon4. is a circle.r " 3 cos !
7. The graph is symmetric about the line andpasses through Matches (a).!r, !" " !3, 3#$2".
! " #$2, 8. The graph is symmetric about the polar axis andpasses through Matches (c).!r, !" " !3, 0".
9. The graph has four leaves. Matches (c). 10. The graph has three leaves. Matches (d).
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Page 76
846 Chapter 9 Topics in Analytic Geometry
11.
Not an equivalent equation
Not an equivalent equation
Polar axis:
Equivalent equation
Pole:
Not an equivalent equation
Not an equivalent equation
Answer: Symmetric with respect to polar axis
r " 14 % 4 cos !
r " 14 $ 4 cos!# $ !"
%r " 14 $ 4 cos !
r " 14 $ 4 cos !
r " 14 $ 4 cos!%!"
r " 14 % 4 cos !
r " 14 $ 4!cos # cos ! $ sin # sin !"
r " 14 $ 4 cos!# % !"
%r " 14 $ 4 cos !
%r " 14 $ 4 cos!%!"! "#2
:
r " 14 $ 4 cos ! 12.
Not an equivalent equation
Not an equivalent equation
Polar axis:
Equivalent equation
Pole:
Not an equivalent equation
Not an equivalent equation
Answer: Symmetric with respect to polar axis
r " %12 cos 3!
r " 12 cos!3!# $ !""
%r " 12 cos 3!
r " 12 cos 3!
r " 12 cos!3!%!""
r " %12 cos 3!
r " 12 cos!3!# % !""
%r " 12 cos 3!
%r " 12 cos!3!%!""! "#
2:
r " 12 cos 3!
13.
Equivalent equation
Polar axis:
Not an equivalent equation
Not an equivalent equation
%r "4
1 $ sin !
%r "4
1 $ sin!# % !"
r "4
1 % sin !
r "4
1 $ sin!%!"
r "4
1 $ sin !
r "4
1 $ sin # cos ! % cos # sin !
r "4
1 $ sin!# % !"! "#2
:
r "4
1 $ sin !
Pole:
Not an equivalent equation
Not an equivalent equation
Answer: Symmetric with respect to ! "#2
r "4
1 % sin !
r "4
1 $ sin!# $ !"
%r "4
1 $ sin !
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Page 77
Section 9.7 Graphs of Polar Equations 847
14.
Not an equivalent equation
Not an equivalent equation
Polar axis:
Equivalent equation
r "2
1 % cos !
r "2
1 % cos!%!"
r "2
1 $ cos !
r "2
1 % !cos # cos ! $ sin # sin !"
r "2
1 % cos!# % !"
%r "2
1 % cos !
%r "2
1 % cos!%!"! "#2
:
r "2
1 % cos !
Pole:
Not an equivalent equation
Not an equivalent equation
Answer: Symmetric with respect to the polar axis
r "2
1 $ cos !
r "2
1 % !cos # cos ! % sin # sin !"
r "2
1 % cos!# $ !"
%r "2
1 % cos !
15.
Equivalent equation
Polar axis:
Not an equivalent equation
Not an equivalent equation
Pole:
Not an equivalent equation
Not an equivalent equation
Answer: Symmetric with respect to ! "#2
r " %6 sin !
r " 6 sin!# $ !"
%r " 6 sin !
%r " 6 sin !
%r " 6!sin # cos ! % cos # sin !"%r " 6 sin!# % !"
r " %6 sin !
r " 6 sin!%!"
r " 6 sin !
%r " 6 sin!%!"! "#2
:
r " 6 sin ! 16.
Equivalent equation
Polar axis:
Equivalent equation
Pole:
Equivalent equation
Answer: Symmetric with respect to
polar axis and pole
! "#2
,
r " 4 cot !
r " 4 cot!# $ !"
r " 4 cot !
%r " 4 cot!%!"
%r " 4 cot!# % !"
r " 4 cot !
%r " 4 cot!%!"! "#2
:
r " 4 csc ! cos ! " 4 cot !
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Page 78
848 Chapter 9 Topics in Analytic Geometry
17.
Not an equivalent equation
Not an equivalent equation
Polar axis:
Not an equivalent equation
Not an equivalent equation
Pole:
Equivalent equation
Answer: Symmetric with respect to pole
r2 " 16 sin 2!
!%r"2 " 16 sin!2!"
r2 " %16 sin 2!
!%r"2 " 16 sin!2!# % !""
r2 " %16 sin 2!
r2 " 16 sin!2!%!""
r2 " %16 sin 2!
r2 " 16 sin!2# % 2!"
r2 " 16 sin!2!# % !""
r2 " %16 sin 2!
!%r"2 " 16 sin!2!%!""! "#2
:
r2 " 16 sin 2! 18.
Equivalent equation
Polar axis:
Equivalent equation
Pole:
Equivalent equation
Answer: Symmetric with respect to
polar axis and pole
! "#2
,
r2 " 25 cos 4!
!%r"2 " 25 cos 4!
r2 " 25 cos 4!
r2 " 25 cos!4!%!""
r2 " 25 cos 4!
!%r"2 " 25 cos!4!%!""! "#2
:
r2 " 25 cos 4!
19.
Maximum: when
! "#2
sin ! " 1
r " 0 when 1 % sin ! " 0
! "3#
2#r# " 20
! "3#
2 or Not possible
sin ! " %1 sin ! " 3
1 % sin ! " 2 or 1 % sin ! " %2
#1 % sin !# " 2
" 10#1 % sin !# " 10!2" " 20
#r# " #10!1 % sin !"# 20.
Maximum:
Zero: r " 0 when ! "2#
3,
4#
3
#r# " 18 when ! " 0
! " 0
cos ! " 1
" 6 $ 12#cos !# " 18
#r# " #6 $ 12 cos !# " #6# $ #12 cos !#
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Page 79
Section 9.7 Graphs of Polar Equations 849
21.
Maximum: when
! "#
6,
#
2,
5#
6
r " 0 when cos 3! " 0
! " 0, #
3,
2#
3, ##r# " 4
! " 0, #
3,
2#
3, #
cos 3! " ±1
#cos 3!# " 1
#r# " #4 cos 3!# " 4 #cos 3!# " 4 22.
Maximum: when
Zero: when ! " 0, #
2, #,
3#
2, 2#r " 0
! "#
4,
3#
4,
5#
4,
7#
4#r# " 1
#r# " #sin 2!#r " sin 2!
23.
Circle
2 4 6 8
!2
0
r " 5 24.
Line
1 20
2!
! " %5#3 25.
Symmetric with respect to
Circle with radius of
21 30
2!
3$2
! " #$2
r " 3 sin !
26.
Circle
Radius: 1, center:
0
!2
1 3
!1, 0"
r " 2 cos ! 27.
Cardioid
2 40
2!
r " 3!1 % cos !" 28.
Cardioid
0
!2
4 6 8
r " 4!1 $ sin !"
29.
Limaçon
01 2
2!
r " 3 % 4 cos ! 30.
Limaçon with inner loop
0
!2
1 2
r " 1 % 2 sin ! 31.
Limaçon
4 6 8
!2
0
r " 4 $ 5 sin !
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Page 80
850 Chapter 9 Topics in Analytic Geometry
32.
Limaçon
4 6 8 10 12
!2
0
r " 3 $ 6 cos ! 33.
Rose curve
01 65
2!
r " 5 cos 3! 34.
Rose curve
01
2!
r " %sin 5!
35.
Rose curve, four petals
04 5 6
2!
r " 7 sin 2! 36.
Rose curve, five petals
01 2 3 4
2!
r " 3 cos 5! 37.
0 " ! < 2#
18"18
"12
12
r " 8 cos 2!
39.
0 " ! < 2#
18"18
"14
10
r " 2!5 % sin !"38.
0 " ! " 2#
"2
2
3"3
40.
0 " ! " 2#
"12
4
12"12
41.
"6
6
4"14
0 " ! " 2#r " 3 % 6 cos !, 42.
0 " ! " 2#
"6
2
6"6
43.
"4
4
6"6
0 " ! "#2
r "3
sin ! % 2 cos !,
44.
0 " ! " 2#
"4
4
6"6
45.
"2
2
3"3
%2# " ! " 2#r2 " 4 cos 2!, 46.
Graph both functions using
"4
4
6"6
0 " ! " 2#.
r " ±3%sin !
r2 " 9 sin !
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Page 81
Section 9.7 Graphs of Polar Equations 851
47.
"2
2
3"3
r " 4 sin ! cos2 !, 0 " ! " # 48.
0 " ! " #
"2
2
3"3
49.
0 " ! < 2#
18"18
"10
14
r " 2 csc ! $ 6
50.
0 " ! " 2#
"12
12
18"18
51.
Answers will vary.
"2000
"1400
400
200
r " e2! 52.
Answers will vary.
#15 30
#15
15
r " e!$2
53.
"4
4
6"6
0 " ! < 2#r " 3 % 2 cos !, 54.
0 " ! < 2#
"7
1
6"6
55.
"2
2
3"3
r " 2 cos&3!
2 ', 0 " ! < 4#
56.
0 " ! < 4#
"4
4
6"6
57.
Use and
"1
1
1"1
r2 " %%sin 2!."r1 " %sin 2!!
r2 " sin 2!, 0 " ! <#
2
58.
r "±1%!
0 < ! < &
"1
1
1.5"1.5
59.
is an asymptote.
"4
4
6"6
x " %1
r " 2 % sec !
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Page 82
852 Chapter 9 Topics in Analytic Geometry
60.
The graph has an asymptote at y " 1.
" ±# yy % 1#%3 $ 2y % y2 x " ±%y2!3 $ 2y % y2"
! y % 1"2
x2 "y2!3 $ 2y % y2"
! y % 1"2
x2 $ y2 "4y2
! y % 1"2
!±%x 2$ y 2" "2y
y % 1
!±%x 2 $ y 2" ! y % 1" " 2y
!±%x2 $ y2"! y" " 2y $ !±%x2 $ y2" r!r sin !" " 2r sin ! $ r
r sin ! " 2 sin ! $ 1
"3
5
6"6
r " 2 $ csc ! " 2 $1
sin !
61.
is an asymptote.y " 2
"1
3
3"3
r "2!
62.
"4
4
6"6
63. True. It has five petals. 64. False. For example, let r " cos 3!.
65. Answers will vary.r " cos!5!" $ n cos !, 0 " ! < #;
n " %5
"4
4
6"6
n " %4
"4
4
6"6
n " %3
"4
4
6"6
n " %2
"2
2
3"3
n " %1
"2
2
3"3
n " 0
"2
2
3"3
n " 1
"2
2
3"3
n " 2
"2
2
3"3
n " 3
"4
4
6"6
n " 4
"4
4
6"6
n " 5
"4
4
6"6 ©H
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Page 83
Section 9.7 Graphs of Polar Equations 853
69. (a)
(b)
(c)
(d)
" 4 sin ! cos !
" 2 sin 2!
" 2 sin!2! % 2#"
r " 2 sin(2!! % #")
" %3 cos 2! % sin 2!
" 2 sin&2! %4#
3 '
r " 2 sin*2&! %2#
3 '+ " %4 sin ! cos !
" %2 sin 2!
" 2 sin!2! % #"
r " 2 sin*2&! %#
2'+ " sin 2! % %3 cos 2!
" 2 sin&2! %#
3'
r " 2 sin*2&! %#
6'+ 70. (a)
(b)
–2 –1 1 2
–3
–2
1
x
y
r " 1 % sin&! %#
4'
–2 1 2
–3
–1
1
x
y
r " 1 % sin !
66. The graph of is rotated about the pole through an angle Let be any point on the graph of Then is rotated through the angle and since
it follows that is on the graph of r " f !! % '".!r, ! $ '"r " f !!! $ '" % '" " f !!",',!r, ! $ '"r " f !!".!r, !"'.r " f!!"
67. Use the result of Exercise 66.
(a) Rotation:
Original graph:
Rotated graph:
(b) Rotation:
Original graph:
Rotated graph:
(c) Rotation:
Original graph:
Rotated graph: r " f &sin&! %3#
2 '' " f !cos !"
r " f !sin !"
' "3#
2
r " f !sin!! % #"" " f !%sin !"
r " f !sin !"
' " #
r " f &sin&! %#
2'' " f !%cos !"
r " f !sin !"
' "#
2
68. (a)
(b)
(c)
(d)
r " 2 % cos !
r " 2 % sin&! %3#
2 'r " 2 $ sin !
r " 2 % sin!! % #"
r " 2 $ cos !
r " 2 % sin&! %#
2'
" 2 %%22
!sin ! % cos !"
r " 2 % sin&! %#
4'
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Page 84
Vocabulary Check1. conic 2. eccentricity, 3. (a) i (b) iii (c) iie
854 Chapter 9 Topics in Analytic Geometry
Section 9.8 Polar Equations of Conics
! The graph of a polar equation of the form
or
is a conic, where is the eccentricity and is the distance between the focus (pole) and the directrix.
(a) If the graph is an ellipse.
(b) If the graph is a parabola.
(c) If the graph is a hyperbola.
! Guidelines for finding polar equations of conics:
(a) Horizontal directrix above the pole:
(b) Horizontal directrix below the pole:
(c) Vertical directrix to the right of the pole:
(d) Vertical directrix to the left of the pole: r !ep
1 " e cos #
r !ep
1 $ e cos #
r !ep
1 " e sin #
r !ep
1 $ e sin #
e > 1,
e ! 1,
e < 1,!p!e > 0
r !ep
1 ± e sin #r !
ep1 ± e cos #
71.
Circle
!4
4
6!6
k ! 0
r ! 2 $ k cos #
Convex limaçon
!4
4
6!6
k ! 1
Cardioid
!4
4
8!4
k ! 2
Limaçon with inner loop
!4
4
8!4
k ! 3
72.
(a)
k ! 1.5: 0 ! # < 4%
!4
4
6!6
r ! 3 sin k #
(b)
k ! 2.5: 0 ! # < 4%
!4
4
6!6
(c) Yes. Answers will vary.
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Page 85
Section 9.8 Polar Equations of Conics 855
3.
(a) Parabola
(b) Ellipse
(c) Hyperbola!8
4
9!9
a
c
br !
2e1 " e sin #
4. (a) Parabola
(b) Ellipse
(c) Hyperbola
!4
8
9!9
ab
c
5.
parabola
Vertical directrix to left of pole
Matches (b).
e ! 1 "
r !4
1 " cos #6.
ellipse
Vertical directrix to left of pole
Matches (c).
e !12
"
r !3
2 " cos #!
3"21 " #1"2$ cos #
8.
hyperbola
Horizontal directrix below the pole.
Matches (e).
e ! 3 "
r !4
1 " 3 sin #9.
hyperbola
Horizontal directrix above the pole.
Matches (d).
e ! 2 "
r !3
1 $ 2 sin #
13.
ellipse
Vertices:
#r, #$ ! %43
, 0&, %45
, %&
e !14
, p ! 4,0
1 2
"2r !
44 " cos #
!1
1 " #1"4$ cos #
1.
(a) Parabola
(b) Ellipse
(c) Hyperbola!4
4
8!4
a
bc
r !2e
1 $ e cos #2. (a) Parabola
(b) Ellipse
(c) Hyperbola
!4
4
4!8
a
bc
7.
ellipse
Vertical directrix to right of pole
Matches (f).
e !12
"
r !3
2 $ cos #!
3"21 $ #1"2$ cos #
10.
parabola
Vertex:
Matches (a).
%2, %2&
e ! 1 "
r !4
1 $ sin #11.
parabola
Vertex: #r, #$ ! #1, %$
e ! 1 "
r !2
1 " cos #12.
Vertex: #1, %"2$
e ! 1 " parabola
r !2
1 $ sin #
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Page 86
856 Chapter 9 Topics in Analytic Geometry
17.
ellipse
Vertices:
%2, %2&, %6,
3%2 &
0
"2
1 2 4 5
e !12
"
r !6
2 $ sin #!
#1"2$#6$1 $ #1"2$ sin #
18.
hyperbola
Vertices: #5, 0$, %"53
, %&e ! 2 "
r !5
"1 $ 2 cos #!
"51 " 2 cos #
19.
hyperbola
Hyperbola
Vertices:
%14
, %&#r, #$ ! %"34
, 0&,0
1 2 3 4 5
"2
e ! 2 "
r !3
4 " 8 cos #!
3"41 " 2 cos #
23.
Hyperbola
!1
1 $ #17"14$ sin #
!3
9
9!9
r !14
14 $ 17 sin #24.
Ellipse
!10
10
15!15
r !12
2 " cos #
14.
Vertices: #r, #$ ! %78
, %2&, %7
6,
3%2 &
e !17
" ellipse
r !7
7 $ sin #!
11 $ #1"7$ sin #
15.
Vertices:
#r, #$ ! %87
, %2&, %8,
3%2 &
01 3 4 5 6
"2e !
34
" ellipse
r !8
4 $ 3 sin #!
21 $ #3"4$ sin #
16.
ellipse
Vertices: #6, 0$, %65
, %&
e !23
"
r !6
3 " 2 cos #!
21 " #2"3$ cos #
20.
hyperbola
Vertices:
#r, #$ ! %56
, %2&, %"
53
, 3%2 &
e ! 3 "
r !10
3 $ 9 sin #!
10"31 $ 3 sin #
21.
Parabola
!4
4
6!6
r !"5
1 " sin #22.
Hyperbola
!2
2
3!3
r !"1
2 $ 4 sin #
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Page 87
Section 9.8 Polar Equations of Conics 857
28.
!2
2
3!3
29.
3#3
#2
2 30.!9
!9
9
3
31.
!8
4
9!9
32.
!7
1
4!8
33.
Vertical directrix to the left of the pole
r !1#1$
1 " 1 cos #!
11 " cos #
e ! 1, x ! "1, p ! 1 34.
Horizontal directrix below the pole
r !1#4$
1 " #1$ sin #!
41 " sin #
e ! 1, y ! "4, p ! 4
35.
Horizontal directrix above the pole
r !#1"2$#1$
1 $ #1"2$ sin #!
12 $ sin #
e !12
, y ! 1, p ! 1 36.
Horizontal directrix below pole
!12
4 " 3 sin #r !
(3"4$41 " #3"4$ sin #
e !34
, y ! "4, p ! 4
37.
Vertical directrix to the right of the pole
r !2#1$
1 $ 2 cos #!
21 $ 2 cos #
e ! 2, x ! 1, p ! 1 38.
Vertical directrix to the left of the pole
r !3"2#1$
1 " #3"2$ cos #!
32 " 3 cos #
e !32
, x ! "1, p ! 1
39. Vertex:
Horizontal directrix below the pole
r !1#2$
1 " 1 sin #!
21 " sin #
%1, "%
2& " e ! 1, p ! 2 40. Parabola, vertex:
Vertical directrix to right of pole
r !ep
1 $ e cos #!
161 $ cos #
#8, 0$e ! 1,
25.
Ellipse
2
2!4
!2
26.
Hyperbola
1!5
!2
2 27.
#!
!
!3
#2 !4
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Page 88
858 Chapter 9 Topics in Analytic Geometry
43. Center:
Vertical directrix to the right of the pole
r !10
3 $ 2 cos #
2 !2p
3 $ 2 cos 0!
2p5
" p ! 5
r !#2"3$p
1 $ #2"3$ cos #!
2p3 $ 2 cos #
#4, %$, c ! 4, a ! 6, e !23
45. Center:
Vertical directrix to left of pole
r !20
3 " 2 cos #
20 !2p
3 " 2 ! 2p " p ! 10
r !#2"3$p
1 " #2"3$ cos #!
2p3 " 2 cos #
#8, 0$, c ! 8, a ! 12, e !ca
!23
44. Center:
Horizontal directrix above the pole
r !8
3 $ sin #
p ! 8
2 !p
3 $ sin#%"2$
r !#1&3$ p
1 $ #1&3$ sin #!
p3 $ sin #
%1, 3%
2 &, c ! 1, a ! 3, e !13
46. Center:
Horizontal directrix below the pole
r !5#9"5$
4 " 5 sin #!
94 " 5 sin #
p !95
1 !5p
4 " 5 sin#3%"2$
r !#5"4$p
1 " #5"4$ sin #!
5p4 " 5 sin #
%5, 3%
2 &, c ! 5, a ! 4, e !54
47. Center:
Horizontal directrix above the pole
Substitute the point rather than
in order to get a directrix between the vertices.
r !5#8"5$
3 $ 5 sin #!
83 $ 5 sin #
p !85
1 !5p
3 $ 5 sin#"3%"2$
%"1, 3%2 &%1,
"3%2 &
r !#5"3$p
1 $ #5"3$ sin #!
5p3 $ 5 sin #
%52
, %2&, c !
52
, a !32
, e !53
48. Center:
Horizontal directrix above the pole
r !8
3 $ 5 sin #
1 !5p
3 $ 5 sin#%"2$ " p !85
r !#5"3$p
1 $ #5"3$ sin #!
5p3 $ 5 sin #
%52
, %2&, c !
52
, a !32
, e !ca
!53
42. Vertex:
Horizontal directrix above pole
r !1#20$
1 $ 1 sin #!
201 $ sin #
%10, %
2& " e ! 1, p ! 2041. Vertex:
Vertical directrix to left of pole
r !1#10$
1 " 1 cos #!
101 " cos #
#5, %$ " e ! 1, p ! 10
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Page 89
Section 9.8 Polar Equations of Conics 859
51.
Perihelion distance:
Aphelion distance:
r ! 92.956 ' 106#1 $ 0.0167$ ' 9.4508 ' 107
r ! 92.956 ' 106#1 " 0.0167$ ' 9.1404 ' 107
'9.2930 ' 107
1 " 0.0167 cos #
r !(1 " #0.0167$2)#92.956 ' 106$
1 " 0.0167 cos #52.
Perihelion distance: miles
Aphelion distance: milesa#1 $ e$ ' 4.3381 ' 107
a#1 " e$ ' 2.8585 ' 107
' 3.4462 ' 107
1 " 0.2056 cos #
r !#1 " 0.20562$#35.983 ' 106$
1 " 0.2056 cos #
a ! 35.983 ' 106, e ! 0.2056
53.
Perihelion:
Aphelion:
r ! 77.841 ' 107#1 $ 0.0484$ ' 8.1609 ' 108 km
r ! 77.841 ' 107#1 " 0.0484$ ' 7.4073 ' 108 km
!7.7659 ' 108
1 " 0.0484 cos #
r !#1 " 0.04842$77.841 ' 107
1 " 0.0484 cos #54.
Perihelion distance: km
Aphelion distance: kma#1 $ e$ ' 1.5041 ' 109
a#1 " e$ ' 1.3494 ' 109
' 1.4225 ' 109
1 " 0.0542 cos #
r !#1 " 0.05422$#142.673 ' 107$
1 " 0.0542 cos #
a ! 142.673 ' 107, e ! 0.0542
55. Neptune
Pluto
(a) Neptune:
Pluto:
(b) Neptune: Perihelion: (c)
Aphelion:
Pluto: Perihelion:
Aphelion:
(d) Yes. Pluto is closer to the sun for just a very short time. Pluto was considered the ninth planet because its mean distance from the sun is larger than that of Neptune.
(e) Although the graphs intersect, the orbits do not, and the planets won’t collide.
5.906 ' 109#1 $ 0.2488$ ' 7.3754 ' 109 km
5.906 ' 109#1 " 0.2488$ ' 4.4366 ' 109 km
4.498 ' 109#1 $ 0.0086$ ' 4.5367 ' 109 km!5 # 109 8 # 109
!7 # 109
7 # 1094.498 ' 109#1 " 0.0086$ ' 4.4593 ' 109 km
r !#1 " 0.24882$5.906 ' 109
1 " 0.2488 cos #!
5.5404 ' 109
1 " 0.2488 cos #
r !#1 " 0.00862$4.498 ' 109
1 " 0.0086 cos #!
4.4977 ' 109
1 " 0.0086 cos #
a ! 5.906 ' 109, e ! 0.2488,
a ! 4.498 ' 109, e ! 0.0086,
49. When
Therefore,
Thus, r !ep
1 " e cos #!
#1 " e2$a1 " e cos #
.
a#1 " e2$ ! ep.
a#1 $ e$#1 " e$ ! ep
a#1 $ e$ !ep
1 " e cos 0
# ! 0, r ! c $ a ! ea $ a ! a#1 $ e$. 50. Minimum distance occurs when
Maximum distance occurs when
r !#1 " e2$a
1 " e cos 0!
#1 " e$#1 $ e$a1 " e
! a#1 $ e$
# ! 0.
r !#1 " e2$a
1 " e cos %!
#1 " e$#1 $ e$a1 $ e
! a#1 " e$
# ! %.
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Page 90
860 Chapter 9 Topics in Analytic Geometry
56. (a) Radius of earth miles. Choose
Vertices:
Thus, Thus,
(b) When and the distance from the surface of the earth to the satellite is
(c) When and distance miles.! 38,370# ! 30(, r ' 42,370
15,029 " 4000 ! 11,029 miles.# ! 60(, r ' 15,029
' 7988.11 " 0.937 cos #
.r !ep
1 " e cos #p !
a#1 " e2$e
' 8525.2.
2a !ep
1 " e cos 0$
ep1 " e cos#%$ !
ep1 " e
$ep
1 $ e!
2ep1 " e2
e !ca
!61,340.565,459.5
' 0.937
c ! 65,459.5 " 4119 ! 61,340.5
a !126,800 $ 4119
2! 65,459.5
#126,800, 0$ and #4119, %$
r !ep
1 " e cos #.' 4000
57.
False. The directrix is below the pole.
r !4
"3 " 3 sin #!
"4"31 $ sin #
58.
False. The graph is not an ellipse.
(It is two ellipses.)
r2 !16
9 " 4 cos%# $%4&
59.
For an ellipse, Hence,
r2 !b2
1 " e2 cos2 #.
r2#1 " e2 cos2 #$ ! b2
"r2e2 cos2 # $ r2 ! b2
"r2%ca&
2
cos2 # $ r2 ! b2, e !ca
"r2c2 cos2 # $ r2a2 ! a2b2
b2 " a2 ! "c2.
r2#b2 " a2$ cos2 # $ r2a2 ! a2b2
r2b2 cos2 # $ r2a2 " r2a2 cos2 # ! a2b2
r2 cos2 #
a2 $r2#1 " cos2 #$
b2 ! 1
r2 cos2 #
a2 $r2 sin2 #
b2 ! 1
x2
a2 $y2
b2 ! 1 60.
!"b2
1 " e2 cos2 #
r2 !b2
e2 cos2 # " 1
r2#e2 cos2 # " 1$ ! b2
r2e2 cos2 # " r2 ! b2
r2%ca&
2 cos2 # " r2 ! b2, e !
ca
r2c2 cos2 # " r2a2 ! a2b2
a2 $ b2 ! c2
r2#b2 $ a2$ cos2 # " r2a2 ! a2b2
r2b2 cos2 # " r2a2 $ r2a2 cos2 # ! a2b2
r2 cos2 #
a2 "r2#1 " cos2 #$
b2 ! 1
r2 cos2 #
a2 "r2 sin2 #
b2 ! 1
x2
a2 "y2
b2 ! 1
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Page 91
Section 9.8 Polar Equations of Conics 861
61.
r2 !144
1 " #25"169$ cos2 #!
24,336169 " 25 cos2 #
a ! 13, b ! 12, c ! 5, e !513
x2
169$
y2
144! 1
63.
r2 !b2
1 " e2 cos2 #!
161 " #9"25$ cos2 #
!400
25 " 9 cos2 #
a ! 5, b ! 4, c ! 3, e !35
x2
25$
y2
16! 1
64.
r2 !"b2
1 " e2 cos2 #!
"41 " #10"9$ cos2 #
!36
10 cos2 # " 9
a ! 6, b ! 2, c ! *40 ! 2 *10, e !2*10
6!
*103
x2
36"
y2
4! 1
62.
r2 !"16
1 " #25"9$ cos2 #!
14425 cos2 # " 9
a ! 3, b ! 4, c ! 5, e !53
x2
9"
y2
16! 1
65. Center: ,
r2 !"b2
1 " e2 cos2 #!
"91 " #25"16$ cos2 #
!144
25 cos2 # " 16
" b ! 3b2 ! c2 " a2 ! 25 " 16 ! 9
c ! 5, a ! 4, e !54
#x, y$ ! #0, 0$
66. Center: ,
r2 !b2
1 " e2 cos2 #!
91 " #16"25$ cos2 #
!225
25 " 16 cos2 #
" b ! 3b2 ! a2 " c2 ! 25 " 16 ! 9
c ! 4, a ! 5, e !45
#x, y$ ! #0, 0$
67.
Vertical directrix to left of pole
(a)
!5
7
9!9
!6
6
9!9
e ! 0.4 " ellipse
r !4
1 " 0.4 cos #
(b)
Vertical directrix to right of pole
Graph is reflected in line
Horizontal directrix below pole
rotation counterclockwise90(
r !4
1 " 0.4 sin #
# ! %"2.
r !4
1 $ 0.4 cos #
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Page 92
862 Chapter 9 Topics in Analytic Geometry
68. The lengths of the major andminor axes increase as pincreases.
Example:
r !#0.5$4
1 $ #0.5$ sin #
r !#0.5$2
1 $ #0.5$ sin #
69. Answers will vary. 70.
Circle
x2 $ y2 ! ay $ bx
r2 ! a r sin # $ b r cos #
r ! a sin # $ b cos #
71.
# !%6
$ n%
tan # !1*3
!*33
4*3 tan # " 3 ! 1 72.
x !%3
$ 2n%, 5%3
$ 2n%
cos x !12
6 cos x " 2 ! 1 73.
# !%3
$ n%, 2%3
$ n%
sin # ! ±*32
sin2 # !34
12 sin2 # ! 9
74.
x !%3
$ n%, 2%3
$ n%
sin x !±*3
2
sin2 x !34
csc2 x !43
9 csc2 x " 10 ! 2 75.
x !%2
$ n%
cot x ! 0
2 cot x ! 0
2 cot x ! 5 cos %2
76.
# !%3
$ 2n%, 5%3
$ 2n%
cos # !12
sec # ! 2
*2 sec # ! 2*2
*2 sec # ! 2 csc %4
For Exercises 77–80: sin v ! "1*2
cos v !1*2
,cos u !45
,sin u ! "35
,
77.
!*210
!1
5*2
!45% 1
*2& " %"35&%"
1*2&
cos#u $ v$ ! cos u cos v " sin u sin v 78.
!"7
5*2!
"7*210
! %"35 &% 1
*2& $ %"1*2&%
45&
sin#u $ v$ ! sin u cos v $ sin v cos u
79.
!*210
!1
5*2
! %"35&% 1
*2& " %"1*2&%
45&
sin#u " v$ ! sin u cos v " sin v cos u 80.
!7
5*2!
7*210
! %45&% 1
*2& $ %"35 &%"1
*2&cos#u " v$ ! cos u cos v $ sin u sin v
81. 12C9 ! 220 83. 10P3 ! 72082. 18C16 ! 153 84. 29P2 ! 812
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Page 93
Review Exercises for Chapter 9
Review Exercises for Chapter 9 863
1.
x2 ! y2 " 25
" !9 ! 16 " !25 " 5
Radius " !"#3 # 0#2 ! "#4 # 0#2 2.
x2 ! y2 " 289
" !64 ! 225 " !289 " 17
Radius " !"8 # 0#2 ! "#15 # 0#2
3.
"x # 2#2 ! " y # 4#2 " 13
Center " $5 ! "#1#2
, 6 ! 2
2 % " "2, 4#
"12!36 ! 16 "
12!52 " !13
Radius "12!"5 # "#1##2 ! "6 # 2#2 4.
"x # 2#2 ! " y ! 1#2 " 32
Center " $#2 ! 62
, 3 # 5
2 % " "2, #1#
"12!64 ! 64 " 4!2
Radius "12!"6 # "#2##2 ! "#5 # 3#2
5.
Center:
Radius: 6
"0, 0#
x2 ! y2 " 36
12
x2 ! 12
y2 " 18 6.
Center:
Radius:2!3
"2!3
3
"0, 0#
x2 ! y2 " 43
34
x2 ! 34
y2 " 1
7.
Center:
Radius: 1
"12, #3
4# "x # 1
2#2! " y ! 3
4#2" 1
16"x # 12#2
! 16" y ! 34#2
" 16
16"x2 # x ! 14# ! 16" y2 ! 3
2 y ! 916# " 3 ! 4 ! 9
16x2 ! 16y2 # 16x ! 24y # 3 " 0
8.
Center:
Radius: 72
"#4, 3#
"x ! 4#2 ! " y # 3#2 " 494
4"x ! 4#2 ! 4" y # 3#2 " 49
4"x2 ! 8x ! 16# ! 4" y2 # 6y ! 9# " #51 ! 64 ! 36
4x2 ! 4y2 ! 32x # 24y ! 51 " 0
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Page 94
864 Chapter 9 Topics In Analytic Geometry
9.
Center:
Radius: 4!1!2!3!4!6!7 2 3
!2!3!4!5!6
!8
2
x
y"#2, #3#
"x ! 2#2 ! " y ! 3#2 " 16
"x2 ! 4x ! 4# ! " y2 ! 6y ! 9# " 3 ! 4 ! 9 10.
Center:
Radius: 7
!10!14 2 4 6
!4!6
2468
10
14
x
y"#4, 5#
"x ! 4#2 ! " y # 5#2 " 49
"x2 ! 8x ! 16# ! " y2 # 10y ! 25# " 8 ! 16 ! 25
11. intercepts:
intercepts:
No interceptsy-
" y ! 1#2 " #2, impossible
"0 # 3#2 ! " y ! 1#2 " 7y-
"3 ± !6, 0# x " 3 ± !6
x # 3 " ±!6
"x # 3#2 " 6
"x # 3#2 ! "0 ! 1#2 " 7x- 12. intercepts:
No intercepts
intercepts:
"0, 6 ± !2 # y " 6 ± !2
y # 6 " ±!2
" y # 6#2 " 2
"0 ! 5#2 ! " y # 6#2 " 27y-
x-
"x ! 5#2 " #9, impossible
"x ! 5#2 ! "0 # 6#2 " 27x-
13.
Vertex:
Focus:
Directrix: x " #1
"1, 0#
"0, 0#
y2 " 4"1#x, p " 1
–2 2 4 6 8 10
–6
–4
–2
2
4
6
x
y 4x # y2 " 0 14.
Vertex:
Focus:
Directrix: y " 2
"0, #2#
"0, 0#
x2 " 4"#2#y, p " #2–12 –9 –6 6 9 12
–21
–18
–15
–12
–9
–6
3
x
y y " #18 x2
15.
Vertex:
Focus:
Directrix: x " 9
"#9, 0#
x
y
!4!8!12!16!20 4!4
4
12
"0, 0#
y2 " #36x " 4"#9#x, p " #9
12 y2 " #18x
12 y2 ! 18x " 0 16.
Vertex:
Focus:
Directrix: y " # 1128
"0, 1128#
x
y
316
18
18
316
14
516
1128
116
18
316
! ! ! (0, 0)
0, ))
"0, 0#
x2 " 132 y " 4" 1
128#y, p " 1128
8x2 " 14 y
14 y # 8x2 " 0
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Page 95
Review Exercises for Chapter 9 865
17. Vertex:
Focus:
Parabola opens to left.
y2 " #24x
y2 " 4"#6#x
y2 " 4px
"#6, 0#
"0, 0# 18. Vertex:
Focus:
Vertical axis,
"x # 4#2 " #8"y # 2#
"x # 4#2 " 4"#2#"y # 2#
p " #2
"4, 0#
"4, 2#
19. Vertex:
Passes through
Vertical axis
"x ! 6#2 " #9"y # 4#
"x ! 6#2 " 4"#94#"y # 4#
#94 " p
36 " #16p
"0 ! 6#2 " 4p"0 # 4#
"x ! 6#2 " 4p"y # 4#
"0, 0#
"#6, 4# 20. Vertex:
on graph:
"y # 5#2 " 4" 2524#x " 25
6 x
"0 # 5#2 " 4p"6# ! p " 2524
"6, 0#
"y # 5#2 " 4p"x # 0# " 4px
"0, 5#
21.
Focus:
Slope of tangent line:
intercept:
d1
d2
!2
!1
F
(0, )b
!2 2x
y
(2, !2)
"1, 0#x-
y " #2x ! 2
Equation: y ! 2 " #2"x # 2#
b ! 20 # 2
"4
#2" #2
b " 2!12
! b "52
!d1 " d2
d2 "!"2 # 0#2 ! $#2 !12%
2"
52
d1 "12
! b
$0, #12%
x2 " #2y " 4$#12%y, p " #
12
22.
Focus:
Let be the intercept of the tangent line.
intercept: "8, 0#x-
y "14
x # 2
y # 0 "14
"x # 8#
m "#4 # 0#8 # 8
"14
12
! b "172
! b " 8
d2 "!$#8 !12%
2! "#4 # 0#2 "
172
d1 "12
! b
x-"b, 0#
$#12
, 0%
p " #12
4$#12
x% " y2
#2x " y2
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Page 96
866 Chapter 9 Topics In Analytic Geometry
23.
on curve:
if width is meters.8!6x2 " 96 ! x " 4!6 !y " 0
y "#x2 ! 96
8
x2 " 4"#2#"y # 12# " #8y ! 96
16 " 4p"10 # 12# " #8p ! p " #2
"4, 10#
x2 " 4p"y # 12#
24. (a) Parabola:
Vertex:
Passes through
Circle:
Passes through
Radius:
Center:
x2 ! "y ! 4!3 #2 " 64
k " #!48 " #4!3
k2 " 48
16 ! k2 " 64
"±4#2 ! "0 # k#2 " 82
x2 ! "y # k#2 " 82
"0, k#
r " 8
"±4, 0#
x2 " #4"y # 4#
#1 " p
16 " #16p
16 " 4p"0 # 4#
x2 " 4p"y # 4#
"±4, 0#
"0, 4#(b) Parabola:
Circle:
d " #14 x2 # !64 # x2 ! 4 ! 4!3
d " "#14 x2 ! 4# # "!64 # x2 # 4!3 #
x2 ! "y ! 4!3#2" 64 ! y " !64 # x2 # 4!3
x2 " #4"y # 4# ! y " #14 x2 ! 4
x 0 1 2 3 4
d 2.928 2.741 2.182 1.262 0
25.
Center:
Vertices:
Foci:
"!32
"2!3
4
Eccentricity "ca
"0, ±2!3 #
1 !1!3!4 3 4
–3
–2
1
2
3
x
y"0, ±4#
"0, 0#
a " 4, b " 2, c " !16 # 4 " !12 " 2!3
x2
4!
y2
16" 1 26.
Center:
Vertices:
Foci:
Eccentricity "ca
"13
"±1, 0#
!1!2!4 1 2 4!1
!2
!4
1
2
4
x
y"±3, 0#
"0, 0#
a " 3, b " 2!2, c " !9 # 8 " 1
x2
9!
y2
8" 1
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Page 97
Review Exercises for Chapter 9 867
27.
Center:
Vertices:
Foci:
Eccentricity "ca
"!33
"4, #4 ± !3 #"4, #1#, "4, #7#
!1!2 1 2 3 5 6 7 8
!2!3!4!5!6!7!8!9
1x
y"4, #4#
a " 3, b " !6, c " !9 # 6 " !3
"x # 4#2
6!
" y ! 4#2
9" 1 28.
Center:
Vertices:
Foci:
Eccentricity "ca
"!10
4
"#1 ± !10, 3 #"3, 3#, "#5, 3#
!1!2!3!4!5!6 1 2 3 4
12
456789
x
y"#1, 3#
a " 4, b " !6, c " !16 # 6 " !10
"x ! 1#2
16!
" y # 3#2
6" 1
29. (a) (c)
(b) Center:
Vertices:
Foci:
e "ca
"!74
"1, #4 ± !7#"1, 0#, "1, #8#
a " 4, b " 3, c " !16 # 9 " !7
"1, #4#
"x # 1#2
9!
"y ! 4#2
16" 1
16"x # 1#2 ! 9"y ! 4#2 " 144 !1!2!3 1 2 3 4 5
!2
!3
!4
!5
!6
!8
x
y 16"x2 # 2x ! 1# ! 9"y2 ! 8y ! 16# " #16 ! 16 ! 144
30. (a) (c)
(b)
Center:
Vertices:
Foci:
e "!21
5
"#2 ± !21, 3#"3, 3#, "#7, 3#
"#2, 3#
a " 5, b " 2, c " !21
"x ! 2#2
25!
"y # 3#2
4" 1
4"x ! 2#2 ! 25"y # 3#2 " 100
x
y
!2!4!6!8 2 4!2
!4
!6
2
4
6
4"x2 ! 4x ! 4# ! 25"y2 # 6y ! 9# " #141 ! 16 ! 225
31. (a)
—CONTINUED—
"x ! 2#2
1&3!
"y # 7#2
1&8" 1
3"x ! 2#2 ! 8"y # 7#2 " 1
3"x2 ! 4x ! 4# ! 8"y2 # 14y ! 49# " #403 ! 12 ! 392
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Page 98
868 Chapter 9 Topics In Analytic Geometry
31. —CONTINUED—
(b) Center: (c)
Vertices:
Foci:
Eccentricity:ca
"!30&12!3&3
"!10
4
$#2 ±!3012
, 7%$#2 ±
!33
, 7%c2 " a2 # b2 "
13
#18
"524
! c "!3012
a "!33
, b "!24
!1!2!3!4
2
4
6
8
x
y"#2, 7#
32. (a)
(c)
x
y
!1 1 2 3 4
!1
!2
!3
!4
1
'x # "5&2#(2
"5&4# !"y ! 3#2
"1&16# " 1
$x #52%
2! 20"y ! 3#2 "
54
$x2 # 5x !254 % ! 20"y2 ! 6y ! 9# " #185 !
254
! 180
x2 ! 20y2 # 5x ! 120y ! 185 " 0
33. Vertices:
Foci:
x2
25!
y2
9" 1
a " 5, c " 4 ! b " 3
"±4, 0#
"±5, 0#
(b)
Center:
Vertices:
Foci:
e "ca
"!19&4!5&2
"!95
10
$52
±!19
4, #3%
$52
±!52
, #3%$5
2, #3%
a "!52
, b "14
, c "!54
#116
"!19
4
34. Vertices:
Passes through
Vertical major axis
Center:
x2
9&2!
y2
36" 1
b2 "368
"92
4b2 " 1 #
19
"89
22
b2 !22
36" 1
x2
b2 !y2
36" 1
"0, 0#, a " 6
"2, 2#
"0, ±6#
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Page 99
Review Exercises for Chapter 9 869
35. Vertices:
Foci:
Horizontal major axis
Center:
"x # 2#2
25!
y2
21" 1
"x # h#2
a2 !"y # k#2
b2 " 1
b " !25 # 4 " !21
a " 5, c " 2,
"2, 0#
"0, 0#, "4, 0#
"#3, 0#, "7, 0# 36. Vertices:
Foci:
Vertical major axis
Center:
"x # 2#2
3!
"y # 2#2
4" 1
"x # h#2
b2 !"y # k#2
a2 " 1
b " !4 # 1 " !3
a " 2, c " 1,
"2, 2#
"2, 1#, "2, 3#
"2, 0#, "2, 4#
37.
The foci should be placed 3 feet on either side ofthe center and have the same height as the pillars.
c " !a2 # b2 " !25 # 16 " 3b " 4,a " 5, 38.
Longest distance: feet
Shortest distance: feet
Foci:
Distance between foci: feet16!2 ) 22.63
"±8!2, 0#c2 " a2 # b2 " 128
2b " 2"14# " 28
2a " 2"18# " 36
x2
324!
y2
196" 1, a " !324 " 18, b " !196 " 14
39.
Adding,Then
e "ca
) 0.0543.
c " 1.5045 $ 109 # 1.427 $ 109 " 0.0775 $ 109
2a " 2.854 $ 109 ! a " 1.427 $ 109.
a ! c " 1.5045 $ 109
a # c " 1.3495 $ 109 40.
x2
1296!
y2
1241.2" 1
b2 " a2 # c2 " 362 # 7.40162 ) 1241.2
e "ca
" 0.2056 ! c " ae " 7.4016
a "722
" 36
41. (a)
(b)
Center:
Vertices:
Foci:
Eccentricity "ca
"32
"0, ±3#
"0, ±2#
"0, 0#
c " !4 ! 5 " 3
a " 2, b " !5,
y2
4#
x2
5" 1
5y2 # 4x2 " 20 (c)
!1!2!3!4!5 1 2 3 4 5
!3!4!5
1
345
x
y
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Page 100
870 Chapter 9 Topics In Analytic Geometry
42. (a)
x2
"9&4# #y2
"9&4# " 1
x2 # y2 "94
(b)
Center:
Vertices:
Foci:
Eccentricity "ca
"3!2&2
3&2" !2
$±3!2
2, 0%
$±32
, 0%"0, 0#
"!92
"3!2
"3!2
2
c "!94
!94
a "32
, b "32
, (c)
!1!3!4!5 1 3 4 5
!2!3!4!5
12345
x
y
43. (a) (c)
(b) Center:
Vertices:
Foci:
Eccentricity:54
"6, #1#, "#4, #1#
"5, #1#, "#3, #1#
a " 4, b " 3, c " 5"1, #1#,
"x # 1#2
16#
"y ! 1#2
9" 1
9"x # 1#2 # 16"y ! 1#2 " 144
!6 !4 4 62
2
4
6
8
!4
!2
!6
!8
x
y 9"x2 # 2x ! 1# # 16"y2 ! 2y ! 1# " 151 ! 9 # 16
44. (a) (c)
(b) Center:
Vertices:
Foci:
Eccentricity:!29
2
"#1, #3 ± !29#"#1, #1#, "#1, #5#
"#1, #3#, a " 2, b " 5, c " !29
"y ! 3#2
4#
"x ! 1#2
25" 1
25"y ! 3#2 # 4"x ! 1#2 " 100
6 8 10!8 !6 4
!10!8!6!4!2
!12!14
42
6
x
y 25"y2 ! 6y ! 9# # 4"x2 ! 2x ! 1# " #121 ! 225 # 4
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Page 101
Review Exercises for Chapter 9 871
46. (a)
(b) Center: (c)
Vertices:
Foci:
Eccentricity: e " !10
"4 ± !10, 4#"4 ± 1, 4#: "3, 4#, "5, 4#
a " 1, b " 3, c " !10
!2 4 8 10 12!2
2
4
6
8
10
12
x
y"4, 4#
"x # 4#2 #"y # 4#2
9" 1
9"x # 4#2 # "y # 4#2 " 9
9"x2 # 8x ! 16# # "y2 # 8y ! 16# " #119 ! 144 # 16 " 9
47.
x2
16#
y2
20" 1
! b " !20 " 2!5
c2 " a2 ! b2 ! 36 " 16 ! b2
a " 4
x2
a2 #y2
b2 " 1 48. Vertices:
Foci:
Vertical transverse axis
Center:
y2
1#
x2
8" 1
b " !9 # 1 " !8a " 1, c " 3,
"0, 0#
"0, ±3#
"0, ±1#
45. (a)
(b) Center: (c)
Vertices:
Foci:
Eccentricity: e "ca
"!505!101
" !5
$#6 ± !5052
, 1%$#6 ± !101
2, 1%
a2 "1012
, b2 " 202, c2 "1012
! 202 "5052
!20!30 10 20 30!10
!20
!30
10
20
30
x
y"#6, 1#
"x ! 6#2
"101&2# #"y # 1#2
202" 1
"y # 1#2 # 4"x ! 6#2 " #202
"y2 # 2y ! 1# # 4"x2 ! 12x ! 36# " #59 ! 1 # 144
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Page 102
872 Chapter 9 Topics In Analytic Geometry
49. Foci:
Center:
Asymptotes:
"x # 4#2
16&5#
y2
64&5" 1
b "8!5
16 " a2 ! "2a#2 " 5a2 ! a "4!5
,
c2 " a2 ! b2
y " ±2"x # 4# ! ba
" 2 ! b " 2a
"4, 0#
"0, 0#, "8, 0# ! c " 4 50. Vertical transverse axis
Center:
5y 2
16#
5"x # 3#2
4" 1
"y # k#2
a2 #"x # h#2
b2 " 1
a2 "165
b2 "45
"2b#2 ! b2 " 4
a2 ! b2 " c2
ab
" 2 ! a " 2b
"3, 0# ! c " 2
51.
72.2 miles north
x " 60 ! y2 " b2$x2
a2 # 1% " "1002 # 46.52#$ 602
46.52 # 1% ) 5211.57 ! y ) 72.2
x2
a2 #y2
b2 " 1
b " !c2 # a2
c " 100
a " 46.5
2a " 93
20406080
100
20 AB
(100, 0)
(60, 0)
x
y
!20
!40!60!80
!100
!60
(!100, 0)
d2 # d1 " 186,000"0.0005#
52. Let the friends be at and you at the origin The sound at is heard 2 seconds after Thus, and Thus, using miles,
the hyperbola is
Now place the center at and determine the second hyperbola.
1. 2.
and
"x # 1#2
"25&64# #y2
"39&64# " 1
b2 " 1 #2564
"3964
c " 1
21
1
A
D
BCx
y
!2 !1
!1
21
1
A
D
BC
y
x!2 !1
!1
2a " DB # AD " 6$11005280% ! a "
58
"1, 0#
x2
"25&576# #y2
"2279&576# " 1.
b2 " c2 # a2 " 2279576 .a " 5
24, c " 22a " CD # BD " 2"11005280# " 5
12.B:CA.C,B
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Page 103
Review Exercises for Chapter 9 873
53.
Ellipse
3"x # 2#2 ! 2"y ! 3#2 " 1
3"x2 # 4x ! 4# ! 2"y2 ! 6y ! 9# " #29 ! 12 ! 18
3x2 ! 2y2 # 12x ! 12y ! 29 " 0
54.
A " C " 4 ! Circle
4x2 ! 4y2 # 4x ! 8y # 11 " 0
55.
Hyperbola
"y ! 1#2
7#
"x ! 1#2
"14&5# " 1
5"x ! 1#2 # 2"y ! 1#2 " #14
5"x2 ! 2x ! 1# # 2"y2 ! 2y ! 1# " #17 ! 5 # 2
5x2 # 2y2 ! 10x # 4y ! 17 " 0 56.
Parabola
A " 0, C " #4, AC " 0
#4y2 ! 5x ! 3y ! 7 " 0
57.
Hyperbola
"x% #2
8#
"y% #2
8" 1
12
"x% #2 #12
"y% #2 " 4
!22
"x% # y% # !22
"x% ! y% # " 4
xy " 4
x "!22
"x% # y% #, y "!22
"x% ! y% #
& "'4
!cot 2& "A # C
B" 0
A " 0, B " 1, C " 0y" x"
!2
!2
!3
2
2 3 4 5
345
x
yxy # 4 " 0
58.
Hyperbola
"x% #2
1&4#
"y% #2
1&6" 1
#4"x% #2 ! 6"y% #2 " #1
12
"x% #2 !12
"y% #2 # x%y% # 5""x% #2 # "y% #2# !12
"x% #2 ! x%y% !12
"y% #2 " #1
*!22
"x% # y% #+2
# 10*!22
"x% # y% #+*!22
"x% ! y% #+ ! *!22
"x% ! y% #+2
! 1 " 0
x "!22
"x% # y% #, y "!22
"x% ! y% #
A " C " 1 ! cot 2& " 0 ! & "'4
2
1
1 2
x"y"
x
yx2 # 10xy ! y2 ! 1 " 0
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Page 104
874 Chapter 9 Topics In Analytic Geometry
59.
Ellipse
"x% #2
3!
"y% #2
2" 1
4"x% #2 ! 6"y% #2 " 12
5*12
"x% #2 # x%y% !12
"y% #2+ # "x% #2 ! "y% #2 ! 5*12
"x% #2 ! x%y% !12
"y% #2+ " 12
5*!22
"x% # y% #+2# 2*!2
2"x% # y% #+*!2
2"x% ! y% #+ ! 5*!2
2"x% ! y% #+2
" 12
5x2 # 2xy ! 5y2 " 12
x "!22
"x% # y% #, y "!22
"x% ! y% #
& "'4
!cot 2& " 0
A " 5, B " #2, C " 5y" x"
!1 1 2 3!2!3!1
!2
!3
1
2
3
x
y5x2 # 2xy ! 5y2 # 12 " 0
60.
Parabola
y% " #4"x% #2 # 8x%
8"x% #2 ! 16x% ! 2y% " 0
2'"x% #2 ! "y% #2 # 2x%y% ( ! 4'"x% #2 # "y% #2( ! 2'"x% #2 ! "y% #2 ! 2x%y% ( ! 7"x% # y% # ! 9"x% ! y% # " 0
! 7!2*!22
"x% # y% #+ ! 9!2*!22
"x% ! y% #+ " 0
4*!22
"x% # y% #+2
! 8*!22
"x% # y% # !22
"x% ! y% #+!4*!22
"x% ! y% #+2
x "!22
"x% # y% #, y "!22
"x% ! y% #x
x"y"
y
!3!4 2 3 4
!4
2
3
4cot 2& "4 # 4
8" 0 ! & "
'4
61. (a)
Parabola
(b)
(c)
!10
!5 2
2
y ""8x # 5# ± !"5 # 8x#2 # 4"16x2 # 10x#
2
y2 ! "5 # 8x#y ! "16x2 # 10x# " 0
B2 # 4AC " "#8#2 # 4"16#"1# " 0 62. (a) Ellipse
(b)
(c)
!4
!6 6
4
y "8x ± !"64x2# # 4"7#"13x2 # 45#
14
7y2 # 8xy ! "13x2 # 45# " 0
B2 # 4AC " #300 ! ©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
Page 105
Review Exercises for Chapter 9 875
63. (a) (c)
Parabola
(b)
y ""2!2 # 2x# ± !"2x # 2!2#2 # 4"x2 ! 2!2x ! 2#
2
y2 ! "2x # 2!2 #y ! "x2 ! 2!2x ! 2# " 0
0!15 0
10B2 # 4AC " "2#2 # 4"1#"1# " 0
64. (a)
(b)
(c)
!4
!6 6
4
y " 5x ± !24x2 # 1
y "10x ± !96x2 # 4
2
y "10x ± !100x2 # 4"x2 ! 1#
2
y2 # 10xy ! "x2 ! 1# " 0
" 96 > 0 ! Hyperbola
B2 # 4AC " 100 # 4
66.
Adding:
If
If impossible.
Answer: "4, 3#, "4, #3#
x " #254 , 9"#25
4 # " 4y2,
y2 " 9 ! y " ±3
4y2 " 9"4# " 36
x " 4:
"x # 4#"4x ! 25# " 0 ! x " 4, #254
4x2 ! 9x # 100 " 0
4x2 ! 9x " 100
9x # 4y2 " 0
4x2 ! 4y2 " 100
65. Adding the equations,
Then:
Solution: "#10, 12#
" y # 12#2 " 0 ! y " 12
y2 # 24y ! 144 " 0
4"100# ! y2 # 560 # 24y ! 304 " 0
x " #10.!24x ! 240 " 0
67.
x
y
!4!8!12 8
!4
4
8
12
16
t 0 1 2 3
x 1 4 7
y 15 11 7 3 #5#1
#2#5#8
#1#2
68.
x
y
!1!2!3 1 2 4 5 6!1
1
2
3
4
5
6
7
8
x " !t, y " 8 # t
t 0 1 2 3 4
x 0 1 2
y 8 7 6 5 4
!3!2
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Page 106
876 Chapter 9 Topics In Analytic Geometry
69.
!20
10 20
20
!1030
30
10
!10
40
x
y
y " 25 "x ! 1# ! 5 " 2
5 x ! 275 , line
t " 15 "x ! 1# !
x " 5t # 1, y " 2t ! 5 70.
!15!20
!20
10 20
20
!10
5!5
15
5
!15
!10x
y
y " 8 # 3"14"x # 1## " #3
4x ! 354
!t " 14"x # 1#
x " 4t ! 1, y " 8 # 3t
71.
4!4 !2 6 8 10 12
12
10
8
6
4
2
!4
x
y
y " 4"x # 2# # 3 " 4x # 11, x " 2
t2 " x # 2 !
x " t2 ! 2, y " 4t2 # 3 72.
!3!4
!4
2 4
4
!2
1!1 3
2
1
!2
!3
3
x
y
y " "14 ex#2
" 116e2x!ex " 4t, t " 1
4 ex
x " ln 4t, y " t2
73.
t " x1&3 ! y "12
x2&3
!1!2!3!4 1
1
2
3
4
5
2 3 4
!2
!3
!1
x
yx " t3, y "12
t2 74.
y "16x2 # 1!t "
4x
!4 12!4
4
20
8!8!12x
yx "4t, y " t2 # 1
75.
!4
!6 6
4
y " x3
t " x3 ! y " t " x3
y " t
x " 3!t 76.
!2
!3 3
2
y " 3!t " 3!x " x1&3
x " t 77.
!4
!6 6
4
y " t "1x
y " t
x "1t
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Page 107
Review Exercises for Chapter 9 877
78.
!4
!6 6
4
y "1t
"1x
x " t 79.
Line
!4
!6 6
4
y " 2"2t# " 2x
y " 4t
x " 2t 80.
!1
!1 5
3
y " 4!x, x " 0
x " "y2#2 " y4, y " 0
t " y2
x " t2, y " !t
81.
3x ! 4y # 11 " 0
y "114
#34
x
!2
!4 8
6 t "x # 1
4 ! y " 2 # 3$x # 1
4 % " 2 #34
x !34
y " 2 # 3t
x " 1 ! 4t
82.
y " "x # 4#2
t " x # 4
!1
!2 10
7x " t ! 4, y " t2 83.
Vertical line: x " 3
y " t
!4
!3 9
4x " 3
84.
y " 2
!1
!6 6
7x " t 85.
x2 ! y2 " 36
x2
36!
y2
36" 1
cos & "x6
, sin & "y6
12!12
!8
8x " 6 cos &, y " 6 sin &
86.
14!4
!4
8"x # 3#2
9!
"y # 2#2
25" 1
cos & "x # 3
3, sin & "
y # 25
x " 3 ! 3 cos &, y " 2 ! 5 sin & 87.
Other answers possible
x " #t, y " #6t ! 2
x " t, y " 6t ! 2
y " 6x ! 2
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Page 108
878 Chapter 9 Topics In Analytic Geometry
88.
Many answers possible
x " #t, y " 10 ! t
x " t, y " 10 # t 89.
Other answers possible
x " t ! 1, y " "t ! 1#2 ! 2 " t2 ! 2t ! 3
x " t, y " t2 ! 2
y " x2 ! 2
90.
Many answers possible
x " #t, y " #2t3 # 5t
x " t, y " 2t3 ! 5t 91.
or x " t, y " 5
y " y1 ! t"y2 # y1# " 5 ! t"0 # 0# " 5
x " x1 ! t"x2 # x1# " 3 ! t"8 # 3# " 5t ! 3
92.
or x " 2, y " t
" #1 ! t"4 # "#1## " #1 ! 5t
y " y1 ! t"y2 # y1#
x " x1 ! t"x2 # x1# " 2 ! t"2 # 2# " 2 93.
y " y1 ! t"y2 # y1# " 6 ! t"0 # 6# " #6t ! 6
" #1 ! t'10 # "#1#( " 11t # 1
x " x1 ! t"x2 # x1#
94.
or x " 5t, y " 12t
y " y1 ! t"y2 # y1# " 0 ! t"6 # 0# " 6t
x " x1 ! t"x2 # x1# " 0 ! t" 52 # 0# " 5
2t 95. is on the curve:
Hence, v0 ) 900.82"2.024# ) 54.23 ft&sec.
16t2 " 3 !0.57"90#
0.82 ! t ) 2.024
4 " 7 ! 0.57* 900.82t+t # 16t2 !
90 " 0.82v0 t ! v0 "90
0.82t
"90, 4#
96. From Exercise 95,
" 7 ! 30.91t # 16t2
y " 7 ! 0.57"54.23#t # 16t2
x " 0.82"54.23#t " 44.47t
v0 " 54.23. 97. From Exercise 96:
The maximum height is approxi-mately 21.9 feet for t ) 0.97.
00 100
25
98. From Exercise 95,seconds.t ) 2.024
99.
$1, #7'4 %, $#1,
5'4 %, $#1, #
3'4 %
01 2 3
1,4#( (
#2
$1, '4% 100.
$5, #4'3 %$#5,
5'3 %, $5,
2'3 %,
0
#2
!5,3#( (!
1 2 3 4 5
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Page 109
Review Exercises for Chapter 9 879
101.
$#2, '6%, $2,
7'6 %, $2, #
5'6 %
01 2 3
#2
# ))!2, 611!
"r, &# " $#2, #11'
6 % 102.
$#1, #'6%, $#1,
11'6 %, $1, #
7'6 %
0
#)( ,
1 2 3
56
#2
1
"r, &# " $1, 5'6 %
103.
$!5, 2'3 %, $#!5, #
'3%, $#!5,
5'3 %
01 2 3
5, !3
4( (
#2
#
$!5, #4'3 % 104.
"r, &# " $!10, #5'4 %, $#!10, #
'4%, $#!10,
7'4 %
01 2 3 4
10,( )4#3
#2
105.
01 2 3
#2
5, !6
7#( (
"x, y# " $#5!3
2,
52%
"r, &# " $5, #7'6 % 106.
"x, y# " $#4 cos 2'3
, #4 sin 2'3 % " "2, #2!3 #
"r, &# " $#4, 2'3 %
0
#2
!4,3
2#( (1 2 3 4 5
107.
"x, y# " "1, !3 #
y " r sin & " 2$!32 % " !3
x " r cos & " 2$12% " 1
01 2 3 4
#2
# ))2, 35!
$2, #5'3 %
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Page 110
880 Chapter 9 Topics In Analytic Geometry
108.
01 2 3
#2
# ))!1, 611
"x, y# " $#!32
, 12%
y " r sin & " #1$#12%
x " r cos & " #1$!32 %
$#1, 11'
6 % 109.
1 2 30
#2
3,43#( (
"x, y# " $3 cos 3'4
, 3 sin 3'4 % " $#3!2
2,
3!22 %
"r, &# " $3, 3'4 %
110. the origin
0,2#( (
0
#2
1 2 3 4 5
"x, y# " "0, 0#
"r, &# " $0, '2%, 111.
!3 3 6 9 12
!12
!9
!6
!3
3
x
y
(0, !9)
$#9, '2%"r, &# " $9,
3'2 %,
"x, y# " "0, #9#
112.
or
(radians)"5, 2.214#, "#5, 5.356#
"5, 126.87(#, "#5, 306.87(#
r " 5, tan & "#43
x
y
!1!3 !2!4 1 32 4!1
!3
!2
!4
1
3
2
4(!3, 4)
"x, y# " "#3, 4# 113.
x
y
(5, !5)
!1!2 1 2 3 4 5 6!1
!2
!3
!4
!5
!6
1
2
$#5!2, 3'4 %"r, &# " $5!2,
7'4 %,
"x, y# " "5, #5#
114.
Third quadrant,
"r, &# " $2!3, 7'6 %, $#2!3,
'6%
r " 2!3!r2 " "#3#2 ! 3 " 12
& "7'6
x
y
!1!3 !2!4 1 32 4!1
!3
!2
!4
1
3
2
4
!3, ! 3( (
"x, y# " "#3, #!3 #
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Page 111
Review Exercises for Chapter 9 881
129.
liney " #!33
x,
tan & "yx
" #1!3
& "5'6
130.
y " !3x
tan & "yx
" !3
& "4'3
119.
r2 " 5 csc & ) sec &
"r cos &#"r sin &# " 5
xy " 5 120.
r2 " #2 sec & csc &
"r cos &#"r sin &# " #2
xy " #2
121.
r2 "1
3 cos2 & ! 1
r2 '3 cos2 & ! 1( " 1
4r2 cos2 & ! r2"1 # cos2 &# " 1
4"r cos  ! "r sin  " 1
4x2 ! y " 1 122.
r2 "1
2 ! sin2 &
r2"2 ! sin2 &# " 1
r2"2"1 # sin2 &# ! 3 sin2 &# " 1
r2"2 cos2 & ! 3 sin2 &# " 1
2"r cos  ! 3"r sin  " 1
2x2 ! 3y2 " 1
123.
Circle
x2 ! y2 " 52 " 25
r " 5 124.
Circle
x2 ! y2 " 144
r " 12 125.
x2 ! y2 " 3x
r2 " 3r cos &
r " 3 cos &
126.
x2 ! y2 " 8y
r2 " 8r sin &
r " 8 sin & 127.
"x2 ! y2#2 # x2 ! y2 " 0
"x2 ! y2#2 " x2 ! y2 # 2y2
r4 " r2 # 2r2 sin2 &
r2 " 1 # 2 sin2 &
r2 " cos 2&
128.
"x2 ! y2#3 " y2
"x2 ! y2#3&2 " y or
r3 " r sin &
r2 " sin &
131. circle
4 620
#2
r " 5, 132. circle
1 2 40
#2
r " 3, 133. axis
2 310
#2
y-& "'2
,
115.
r " 3
r2 " 9
x2 ! y2 " 9 116.
r " 2!5
r2 " 20
x2 ! y2 " 20 117.
r " 4 cos &
r2 # 4r cos & " 0
x2 ! y2 # 4x " 0 118.
r " 6 sin &
r2 " 6r sin &
x2 ! y2 " 6y
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Page 112
882 Chapter 9 Topics In Analytic Geometry
134. line
1 4320
#2
& " #5'6
, 135. circle
064321
#2
r " 5 cos &, 136. circle
1 320
#2
r " 2 sin &,
137.
Dimpled limaçon
Symmetric with respect to polar axis
is maximum at
(No zeros)r * 0
"r, &# " "9, 0#& " 0:r 0
#2
2 4 6 8 1210
r " 5 ! 4 cos &
138.
Limaçon with inner loop
Symmetric with respect to
is a maximum at
when & ) 3.394, 6.031!sin & " #14
!4 sin & " #1r " 0
& "'2
: $5, '2%,r,
& "'2
01 2 3 4
#2
r " 1 ! 4 sin &
139.
Limaçon with loop
Symmetry: line
Maximum value: when
Zeros: when $sin & "35%& ) 0.6435, 2.4981r " 0
& "3'2,r, " 8,r,-
& "'2
01 2 3 5
#2
r " 3 # 5 sin &
140.
Limaçon with inner loop
Symmetry: polar axis
Maximum: when
Zero: when cos & " 13 ! & ) 1.231, 5.052r " 0
& " ',r, " 80
#2
2
r " 2 # 6 cos &
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Page 113
Review Exercises for Chapter 9 883
141.
Four-leaved rose
Symmetric with respect to polar axis, and pole
The value of is a maximum (3) at
for & "'4
, 3'4
, 5'4
, 7'4
r " 0
'2
, ', 3'2
.& " 0,,r,
& "'2
,0
4
#2
r " #3 cos 2&, 0 # & # 2'
142.
Five-leaved rose
Symmetric with respect to polar axis
is maximum value of 1 at
for & "'10
!2n'10
, n " 0, 1, 2, . . .r " 0
& "n'5
, n " 0, 1, 2, . . .,r,0
1
#2
r " cos 5&
143.
Lemniscate
Symmetry with respect to pole
Maximum value: when
Zeros: when & " 0, '2
, ', 3'2
r " 0
& "'4
, 5'4
!5,r,-0
1 2 3
#2
r2 " 5 sin 2&
144.
Lemniscate
Symmetry: Pole, polar axis, and line
Maximum: when
Zeros: when & "'4
, 3'4
, 5'4
, 7'4
r " 0
& " 0, ', 2',r, " 1
& "'2 0
1
#2
r2 " cos 2&
145.
Parabola
!2
6
6!6
e " 1
r "2
1 # sin &146.
Hyperbola symmetric with and havingvertices at and
!1
3
3!3
"#1, 3'&2#"1&3, '&2#& " '&2
r "1
1 ! 2 sin &, e " 2
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Page 114
884 Chapter 9 Topics In Analytic Geometry
147.
Ellipse
e "35
"4&5
1 # "3&5# cos &!2
2
3!3
r "4
5 # 3 cos &148.
Hyperbola
!4
4
6!6
"e " 4#
r "6
#1 ! 4 cos &"
#61 # 4 cos &
149.
Ellipse
e "13
"5&6
1 ! "1&3# sin &!2
2
3!3
r "5
6 ! 2 sin &150.
Parabola
!4
4
8!4
"e " 1#
r "3
4 # 4 cos &"
3&41 # cos &
151.
Vertical directrix: x " #4
r "4
1 # cos &
e " 1 152. Parabola:
Vertex:
Focus:
r "4
1 ! sin &
"0, 0# ! p " 4
$2, '
2%
r "ep
1 ! e sin &, e " 1
153. Ellipse:
Vertices:
One focus:
r ""2&3#"5&2#
1 # "2&3# cos &"
5&31 # "2&3# cos &
"5
3 # 2 cos &
5 "2&3 p
1 # "2&3# cos 0 ! p "
52
e "ca
"23
,
"0, 0# ! c " 2
"5, 0#, "1, '# ! a " 3
r "ep
1 # e cos &
154. Hyperbola:
Vertices:
One focus:
"7
3 ! 4 cos &"
7&31 ! "4&3# cos &
r ""4&3#"7&4#
1 ! "4&3# cos &
1 "4&3 p
1 ! "4&3# cos 0 ! p "
74
e "ca
"43
,
"0, 0# ! c " 4
"1, 0#, "7, 0# ! a " 3
r "ep
1 ! e cos &
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Page 115
Review Exercises for Chapter 9 885
155.
Use
Perihelion: astronomical units
Aphelion: astronomical units1.512
1 # 0.093) 1.667
1.5121 ! 0.093
) 1.383
r "1.512
1 # 0.093 cos &
2a "0.093p
1 # 0.093 cos 0!
0.093p1 # 0.093 cos '
" 0.1876p " 3.05 ! p ) 16.258, ep ) 1.512
r "ep
1 # e cos &.
e " 0.093
156. Use (horizontal directrix below pole).
(parabola)
When
When distance is approximately 6,430,781 miles.& " #'3
,
r "12,000,0001 # sin &
r "p
1 # sin$#'2 %
"p2
" 6,000,000 ! p " 12,000,000
& "#'
2, r " 6,000,000.
e " 1
r "ep
1 # e sin &
157. False. The term is notsecond degree.
y4- 158. False. There are many setspossible. For example,
x " 3t, y " 3 # 6t.
x " t, y " 3 # 2t
159. (a) Vertical translation
(b) Horizontal translation
(c) Reflection in the axis
(d) Parabola opens more slowly.
y-
160. (a) Major axis horizontal
(b) Circle
(c) Ellipse is flatter.
(d) Horizontal translation
161. The number must be less than 5. The ellipsebecomes more circular and approaches a circle of radius 5.
b
162. The orientation of the graph would be reversed. 163. (a) The speed would double.
(b) The elliptical orbit would be flatter. The lengthof the major axis is greater.
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Page 116
886 Chapter 9 Topics In Analytic Geometry
Chapter 9 Practice Test
1. Find the vertex, focus and directrix of the parabola x2 ! 6x ! 4y " 1 # 0.
2. Find an equation of the parabola with its vertex at and focus at !2, !6".!2, !5"
3. Find the center, foci, vertices, and eccentricity of the ellipse x2 " 4y2 ! 2x " 32y " 61 # 0.
4. Find an equation of the ellipse with vertices and eccentricity e # 12.!0, ±6"
5. Find the center, vertices, foci, and asymptotes of the hyperbola 16y2 ! x2 ! 6x ! 128y " 231 # 0.
6. Find an equation of the hyperbola with vertices at and foci at !±5, 2".!±3, 2"
7. Rotate the axes to eliminate the term. Sketch the graph of the resulting equation, showing both sets of axes.
5x2 " 2xy " 5y2 ! 10 # 0
xy-
8. Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or hyperbola.
(a) (b) x2 " 4xy " 4y2 ! x ! y " 17 # 06x2 ! 2xy " y2 # 0
For Exercises 9 and 10, eliminate the parameter and write the corresponding rectangular equation.
9. x # 3 ! 2 sin $, y # 1 " 5 cos $ 10. x # e2t, y # e4t
11. Convert the polar point to rectangular coordinates.!#2, !3%"$4"
12. Convert the rectangular point to polar coordinates.!#3, !1"
13. Convert the rectangular equation to polar form.4x ! 3y # 12
14. Convert the polar equation to rectangular form.r # 5 cos $
15. Sketch the graph of r # 1 ! cos $.
16. Sketch the graph of r # 5 sin 2$.
17. Sketch the graph of r #3
6 ! cos $.
18. Find a polar equation of the parabola with its vertex at and focus at !0, 0".!6, %$2"
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