9.5 Notes – Hyperbolas
Hyperbolas: the set of all points for which the difference of the distances to two foci is a constant.
d1
d2 – d1= constant
(x, y)
focus focus
d2
center
The imaginary line between the focal points is the ‘transverse’ axis of the hyperbola.
transverse
asymptote
focus focus
(c, 0)
ca
vertex
Horizontal transverse axis
asymptotefocus
(c, 0)
vertex ca
Vertical transverse axis
A hyperbola can be graphed by locating the vertices (using the a distance from the center) and drawing the two asymptotes through the center of the hyperbola. The foci can be located by using the formula: .
c 2 a2 b2
Standard Form of equation for a hyperbola(note the a2 is always in the lead term)
Horizontal Transverse axis Vertical Transverse axis
Asymptote: Asymptote:
Foci:
ba b
a
(x h)2
a2 (y k)2
b2 1
(y k)2
a2 (x h)2
b2 1
c 2 a2 b2
by x h k
a a
y x h kb
Ex 1: State if the hyperbola is horizontal/vertical, find the center, and the eqn of asymptotes.
x 2
9y 2
161
a)
horizontal
Center: (0, 0)4
3y x
by x h k
a
40 0
3y x
Ex 1: State if the hyperbola is horizontal/vertical, find the center, and the eqn of asymptotes.
b)
vertical
Center: (-2, 1)
(y 1)2
49
(x 2)2
91
ay x h k
b
72 1
3y x
7 17 7 11
3 3 3 3y x y x
7 141
3 3y x
7 141
3 3y x
7 72 1 2 1
3 3y x y x
Ex 2: Graph each hyperbola by filling in the missing information
2 2
14 25
x y a)
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
0 0y = 0
2 0 -2 0
c 2 a2 b22 2
14 25
x y
2 4 25c 2 29c
29 5.4c
Ex 2: Graph each hyperbola by filling in the missing information
2 2
14 25
x y a)
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
0 0y = 0
5.45.4 0 -5.4 0
2 0 -2 0
2 2
14 25
x y b
y x h ka
250 0
4y x
25
4y x
Ex 2: Graph each hyperbola by filling in the missing information
2 2
14 25
x y a)
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
0 0y = 0
5.4-5.4 0 5.4 0
25
4y x
2 0 -2 0
2 22 36 2 36y x
Ex 2: Graph each hyperbola by filling in the missing information
2 22 36 2 36y x
2 22 2
136 1
y x
36 36 36
Ex 2: Graph each hyperbola by filling in the missing information
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
-2 2x = -2
-2 8 -2 -4
2 22 2
136 1
y x
c 2 a2 b2
2 36 1c 2 37c
37 6.1c
2 22 2
136 1
y x
Ex 2: Graph each hyperbola by filling in the missing information
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
-2 2x = -2
-2 8 -2 -4
2 22 2
136 1
y x
6.1-2 8.1 -2 -4.1
ay x h k
b
62 2
1y x
2 22 2
136 1
y x
6 2 2y x 6 2 2y x
6 12 2y x 6 12 2y x
6 14y x 6 10y x
Ex 2: Graph each hyperbola by filling in the missing information
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
-2 2x = -2
-2 8 -2 -4
2 22 2
136 1
y x
6.1-2 8.1 -2 -4.1
6 14y x 6 10y x
Ex 3: Write the equation of the hyperbola centered at the origin with foci (-4, 0) (4, 0) and vertices (-3, 0) and (3, 0)
(x h)2
a2 (y k)2
b2 1
2 2
2
( 0) ( 0)1
9
x y
b
c 2 a2 b2
2 2 24 3 b 216 9 b
27 b
2 2
19 7
x y
Ex 4: Write the equation of the hyperbola centered at the origin with foci (0, 2) (0, -2) and vertices (0, 1) and (0, -1)
2 2
2 2
( ) ( )1
y k x h
a b
2 2
2
( 0) ( 0)1
1
y x
b
c 2 a2 b2
2 2 22 1 b 24 1 b
23 b
2 2
11 3
y x
Ex 5: Write the eqn of the hyperbola with center (-2, 1), vertices at (-2, 5) and (-2, -3) and a b-value of 8.
2 2
2 2
( ) ( )1
y k x h
a b
2 2( 1) ( 2)1
16 64
y x
(-2, 1)
(-2, 5)
(-2, -3)
Ex 6: Write the equation in standard form: 2 29 8 54 56 0x y x y
2 28 9 54 56x x y y
2 28 ___ 9 6 ___ 56 ___ ___x x y y 16 –169 81
2 24 9 3 9x y
2 24 3
19 1
x y
2 23 4
11 9
y x