Conics

directrix

axis

a.c = b. V( )c. F( ) d. x2 or y2

e. directrix _________ f. axis _____________

g. equation:___________________

2, 3

2, 4

b. V( )focus

vertex

F( )

“c” is the distance from the vertex to the focus.

x2 opens up or downy2 opens right or left

y = 2x = 2

There are “c” units from the directrix to the vertex.

directrixdirectrix

1

The axis is the line that goes through the vertex and focus.

axis

axis

General form for x2 parabola:y = (x – h)2 + k 1

4c

y = ¼(x – 2)2 + 3

(h, k)

directrix

axis

a.c = b. V( )c. F( ) d. x2 or y2

e. directrix _________ f. axis _____________

g. equation:___________________

-3 -1, 2

-1, -1

y = 5

x = -1 y = -1/12(x + 1)2 + 2

directrix

axis

F( )

V( )

directrix

axis

a.c = b. V( )c. F( ) d. x2 or y2

e. directrix _________ f. axis _____________

g. equation:___________________

2 -1, 4

1, 4

x = -3

y = 4 x = 1/8(y - 4)2 - 1

directrix

axis

F( )

V( )

General form for y2 parabola:x = (y – k)2 + h 1

4c

(h, k)

a.c =___ b. V( )c. F(-2, 0) d. x2 or y2

e. directrix x = -6 f. axis _____________

g. equation:___________________

2 -4, 0

y = 0 x = 1/8(y - 0)2 - 4

directrix

directrix

axis

axis

F( )

V( )

a.c =___ b. V(1, 4)c. F(1, 7) d. x2 or y2

e. directrix _________ f. axis _____________

g. equation:___________________

3

y = 1

y = 1/12(x - 1)2 + 4

directrix

directrix

axis

axis

F( )

V( )

x = 1

a.c =___ b. V( )c. F(-3, -2) d. x2 or y2

e. directrix x = 3 f. axis _____________

g. equation:___________________

-3 0, -2

y = -2 x = -1/12(y + 2)2 + 0

directrix

directrix

axisaxis

F( )V( )

Circles

General form:

(x - h)² + (y - k)² = r²h k rCenter (h, k) radius = r

(x - h)² + (y - k)² = r²Using the form:

Given: Center and radius

(x - )² + (y - )² = ²

Ex. 1: C(5, 2) r = 7

5 2 7

5 2 7

(x - 5)² + (y - 2)² = 49

h k

Ex. 2: C(-3, 4) r = 2 5

(x - h)² + (y - k)² = r²

(x - )² + (y - )² = ²-3 4

-3 4

(x + 3)² + (y - 4)² = 20

h k 2 5

2 5

(x - h)² + (y - k)² = r²Given: Center & Another PointEx. 3: C(4, -7) & (5,

3)(x - )² + (y - )² = ²4 -74 -7h k

To find r2, you can plug in the point or

use the distance formula

( - 4)² + ( + 7)² = r²(1)² + (10)² = r² 101 =

r²(x - 4)² + (y + 7)² = 101

5 3

5 3

To find r2, you can plug in the point or

use the distance formula

(x - h)² + (y - k)² = r²Ex. 4: C origin & (-5, 2)(x - )² + (y - )² = ²0 0

x² + y² = 29

2 22 1 2 1d (x x ) (y y )

2 2d ( 5 0) (2 0)2 2d ( 5) (2) 25 4 29

(x - 0)² + (y - 0)² = ²29

1st Find the center using the midpoint formula:

1 21 2 y yx x ,2 2

Given: Endpoints of diameterEx. 5: (2, 8) & (-4, 6)

are endpoints of the diameter.2 4 8 6,2 2

= (-1, 7)

C =

Then…choose either endpoint and finish like

before.

Let’s useC =(-1, 7) and (2, 8)

C =(-1, 7) and (2, 8)(x - )² + (y - )² = ²-1 7

-1 7h k

( + 1)² + ( - 7)² = r²

2 8

2 8(3)² + (1)² = r²

10 = r²(x + 1)² + (y - 7)² = 10

Remember how to

If a quadratic equation isn’t in

?!?

you will need to

to get it in the correct form.

x2 + y2 + 16x – 22y – 20 = 0

Rewrite the problem:

Here’s how to do it:

x2 + 16x + ( ) +

y2 – 22y + ( ) =

20 +( ) + ( )

Group your x’s and leave a space.Group your y’s and leave a space.Move the constant and leave 2 spaces.

(x + 8)2 + (y – 11)2 = 205

x2 + 16x +( ) + y2 – 22y +( ) = 20 +( ) +( )

Complete the squareHalf the linear term and square it.

Add to both sides.Do this for both x and y.

Factor and simplify.

8 2 6411 2 121

Center (-8, 11) radius = 205

x2 - 12x +( ) + y2 + 8y +( ) = -32 +( ) +( )

x2 + y2 - 12x + 8y + 32 = 0

(x - 6)2 + (y + 4)2 = 20

6 2 364 2 16

Center (6, -4) radius = 2 5

Now you try it:

Ex. 1: (x)² + (y)² = 36 Center (0, 0) radius = 6

Center (0, 0)

left 6down 6

up 6

right 6

Ex. 2: (x - 3)² + (y - 4)² = 25Center (3, 4) radius = 5

Center (3, 4)

right 5left 5 up

5down 5

Ex. 3: (x - 5)² + (y +4)² = 41 Center (5, -4) radius = = 6.441

Center (5, -4)right 6.4left 6.4

down 6.4

up 6

.4

name of ellipse:

center:a:

b:major axis:

minor axis:

vertices:

foci:

name of ellipse:

(0, 0)5 4108(0, 5), (0, -5),(4, 0), (-4, 0)(0, 3), (0, -3)

verticalcenter:

foci:

vertices:

a:

center (0, 0)

focus (0, 3)

focus (0, -3)

x2 + y2 = 116 25x2 + y2 = 116 25

2bc2 = a2 – b2

2aSquare root of the larger denominator.a was under the

y2, so you move a units from the center in a y direction.

b:major axis:minor axis:

b was under the x2, so you move b units from the center in a x direction.

Square root of the smaller denominator.

name of ellipse:

center:a:

b:major axis:

minor axis:

vertices:

foci:

x2 + y2 = 19 20

center: (0, 0)

center (0, 0)

2√5a:

34√56

foci:

vertices:

b:

minor axis: (0, ±2√5)

(±3, 0)(0, ±√11)

name of ellipse:vertical

major axis:

Where is the center of this ellipse?

__ + __ = 1x2 y2

How many units from the center to the curve in an “x” direction?

3

9

How many units from the center to the curve in an “y” direction?

25

5

__ + __ = 1x2 y2

9 25

Where is the center of this ellipse?

__ + __ = 1x2 y2

How many units from the center to the curve in an “x” direction?

4

36

How many units from the center to the curve in an “y” direction?

16

6

__ + __ = 1x2 y2

36 16

x2 + 10y2 = 101

010

10Divide to make the constant

1.

x2 + y2 = 1 10 1

SF:center:

vertices:

foci:

SF:center: (0,

0)

foci:

vertices: (±√10, 0)(0, ±1)(±3, 0)

24x2 + 3y2 = 7272

72

72Divide to make the constant

1.

x2 + y2 = 1 3 24

SF:center:

vertices:

foci:

SF:center: (0,

0)

foci:

vertices: (0, ±2√6)(±√3, 0)(0, ±√21)

x2 - y2 = 1 9 16 x2 - y2 = 1 9 16

center:a: b:vertices

:

foci:

center: (0, 0)

foci:

vertices: (3, 0) (-3, 0)

(5, 0) (-5, 0)

a: b:

3 4“a” is the square root of the positive variable.

“b” is the square root of the negative variable.Will go in the direction

of the positive variable.c2 = a2 +

b2

4(y + 1)2 – 25(x – 3)2 = 1004(y + 1)2 – 25(x – 3)2 = 100

center:a: b:vertices

:

foci:

center: (3, -1)

foci:

vertices: (3, -6) (3, 4)

(3, -1±√29)

a: b:

5 2

Divide each term by 100 to get into form.

100 100 100 (y + 1)2 – (x – 3)2 = 1 25 4 (y + 1)2 – (x – 3)2 = 1 25 4

16x2 - 9y2 + 54y + 63 = 0Getting it into Standard Form

Factor the –9 out of the “y” terms.

Remember: Put the –9 on the right too.

16x2 + (-9y2 + 54y + ( )) = -63 + ( )

16x2 + -9(y2 - 6y + ( )) = -63 + -9( ) 16x2 + -9(y - 3)2 = -

144Divide each term by -144.

(y -3)2 – x2 = 1 16 9 Why did the x and

y terms trade places?

-144 –144 -144

32 9Note: The +54y becomes -6y

(y – 3)2 - x2 = 1 16 9 (y – 3)2 - x2 = 1 16 9

center:a: b:vertices

:

foci:

center: (0, 3)

foci:

vertices: (0, 7) (0, -1)

(0, 8) (0, -2)

a: b:

4 3

9x2 - 4y2 + 54x + 8y + 41 = 0

(9x2+54x+( ))+(-4y2+8y+( )) = -41+ ( ) + ( ) 9(x2+6x+( )) + -4(y2-2y+( )) = -41+ 9( ) + -4( )

9(x + 3)2 – 4(y - 1)2 = 36

(x + 3)2 – (y – 1)2 = 1 4 9

36 36 36

32 912 1

(x + 3)2 – (y – 1)2 = 1 4 9

(x + 3)2 – (y – 1)2 = 1 4 9

center:a: b:vertices

:

foci:

center: (-3, 1)

foci:

vertices: (-5, 1) (-1, 1)(-3±√13, 1)

a: b:

2 3