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Parabolas

Circles

Conics

Circles

Ellipses

Hyperbolas

Circles

General form (x - h)² (y - k)² r²

h

k

r

Center (h, k) radius r

Given Center and radius

k

h

Ex. 1 C(5, 2) r 7

5

2

7

(x - )² (y - )² ²

5

2

7

(x - 5)² (y - 2)² 49

(x - h)² (y - k)² r²

h

k

-3

4

(x - )² (y - )² ²

-3

4

(x 3)² (y - 4)² 20

(x - h)² (y - k)² r²

Given Center Another Point

h

k

-7

Ex. 3 C(4, -7) (5, 3)

4

5

3

(x - )² (y - )² ²

-7

4

( - 4)² ( 7)² r²

5

3

To find r2, you can plug in the point or

use the distance formula

(1)² (10)² r²

101 r²

(x - 4)² (y 7)² 101

(x - h)² (y - k)² r²

Ex. 4 C origin (-5, 2)

(x - )² (y - )² ²

0

0

To find r2, you can plug in the point or

use the distance formula

x² y² 29

Given Endpoints of diameter

Ex. 5 (2, 8) (-4, 6) are endpoints of the

diameter.

(-1, 7)

C

Lets use C (-1, 7) and (2, 8)

Thenchoose either endpoint and finish like

before.

k

h

C (-1, 7) and (2, 8)

-1

7

2

8

(x - )² (y - )² ²

-1

7

( 1)² ( - 7)² r²

2

8

(3)² (1)² r²

10 r²

(x 1)² (y - 7)² 10

More Circle Fun Next Time!!

Circles Day 2

Putting into standard form and graphing.

Remember how to

Complete the Square

?!?

If a quadratic equation isnt in

Standard form for a Circle (x - h)² (y - k)²

r²

you will need to

Complete the Square

to get it in the correct form.

Heres how to do it

x2 y2 16x 22y 20 0

x2 16x ( )

y2 22y ( )

20 ( ) ( )

- Rewrite the problem

- Group your xs and leave a space.

- Group your ys and leave a space.

- Move the constant and leave 2 spaces.

x2 16x ( ) y2 22y ( ) 20 ( )

( )

8 2

64

(x 8)2 (y 11)2 205

Complete the square

- Half the linear term and square it.

- Add to both sides.

- Do this for both x and y.

- Factor and simplify.

Now you try it

x2 y2 - 12x 8y 32 0

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

)

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

Graphing Circles

Ex. 1 (x)² (y)² 36 Center (0, 0)

radius 6

Center (0, 0)

Ex. 2 (x - 3)² (y - 4)² 25 Center (3, 4)

radius 5

(No Transcript)

That's all for circles!!

Parabolas

Circles

Conics

Ellipses

Ellipses

Hyperbolas

Ellipses

(0, 0)

name of ellipse

center

vertical

4

b

a

5

b was under the x2, so you move b units from the

center in a x direction.

major axis

10

focus (0, 3)

minor axis

8

a was under the y2, so you move a units from the

center in a y direction.

(0, 5), (0, -5), (4, 0), (-4, 0)

center (0, 0)

vertices

focus (0, -3)

(0, 3), (0, -3)

foci

x2 y2 1 9 20

(0, 0)

name of ellipse

center

vertical

3

b

a

2v5

major axis

4v5

minor axis

6

center (0, 0)

(0, 2v5) (3, 0)

vertices

foci

(0, v11)

__ __ 1

25

9

5

3

Where is the center of this ellipse?

How many units from the center to the curve in an

x direction?

How many units from the center to the curve in an

y direction?

__ __ 1

36

16

4

6

Where is the center of this ellipse?

How many units from the center to the curve in an

x direction?

How many units from the center to the curve in an

y direction?

x2 y2 1 10 1

x2 10y2 10

SF center vertices foci

SF

(0, 0)

center

(v10, 0) (0, 1)

vertices

foci

(3, 0)

x2 y2 1 3 24

24x2 3y2 72

SF center vertices foci

SF

(0, 0)

center

(0, 2v6) (v3, 0)

vertices

foci

(0, v21)

bye-bye

ellipses

ellipses

ellipses

ellipses

ellipses

ellipses

ellipses

Parabolas

Circles

Conics

Hyperbolas

Ellipses

Hyperbolas

x2 - y2 1 9 16

x2 - y2 1 9 16

Hyperbolas

(0, 0)

center a b vertices foci

center

4

a

b

3

vertices

(3, 0) (-3, 0)

foci

(5, 0) (-5, 0)

Divide each term by 100 to get into form.

(3, -1)

center a b vertices foci

center

2

a

b

5

vertices

(3, -6) (3, 4)

(3, -1v29)

foci

Getting it into Standard Form

16x2 - 9y2 54y 63 0

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

Factor the 9 out of the y terms.

16x2 -9(y2 - 6y ( )) -63 -9( )

Remember Put the 9 on the right too.

16x2 -9(y - 3)2 -144

Divide each term by -144.

(y -3)2 x2 1 16 9

Why did the x and y terms trade places?

(0, 3)

center a b vertices foci

center

3

a

b

4

vertices

(0, 7) (0, -1)

foci

(0, 8) (0, -2)

9x2 - 4y2 54x 8y 41 0

(9x254x( ))(-4y28y( )) -41 ( ) ( )

9(x26x( )) -4(y2-2y( )) -41 9( )

-4( )

9(x 3)2 4(y - 1)2 36

(x 3)2 (y 1)2 1 4 9

(-3, 1)

center a b vertices foci

center

3

a

b

2

vertices

(-5, 1) (-1, 1)

foci

(-3v13, 1)

(No Transcript)

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