Circle Ellipse

Hyperbola Parabola

Conic Sections

Circles x2 + y2 = 16center: (0,0)

radius: 4

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

44

4

4

x2 + y2 = 36

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

center: (0,0)

radius: 6

6

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

x2 + y2 = 60center: (0,0)

radius:

7.746

2 15

60

-6

-4

-2

0

2

-1 1 3 5 7 9

(x-5)2 + (y+2)2 = 9

center: (5,-2)

radius: 3

-6

-4

-2

0

2

-1 1 3 5 7 93

Find the center and radius of the circle: x2 + y2 + 10x – 6y + 33 = 0

x2 + 10x + y2 – 6y = -33Fill in the blanks by completing the square

+ 25 + 9

If you add 25 and 9 to one side, you must add 25 and 9 to the other side

+ 25 + 9

Factor the left hand side

(x + 5)2 + (y – 3)2 = 1

Center: (-5, 3)

Radius: 1

Find the center and radius of the circle: x2 + y2 + 8x – 2y + 13 = 0

x2 + 8x + y2 – 2y = -13Fill in the blanks by completing the square

+ 16 + 1

If you add 16 and 1 to one side, you must add 16 and 1 to the other side

+ 16 + 1

Factor the left hand side

(x + 4)2 + (y – 1)2 = 4

Center: (-4, 1)

Radius: 2

-3

-2

-1

0

1

2

3

-8 -6 -4 -2 0 2 4 6 8

Ellipse

Center = (0,0)Major Axis = 10

Minor Axis = 4

x2 y2

25 4 + = 1

5 52

2

x2 y2

9 36 + = 1Center = (0,0)Major Axis =12Minor Axis = 6

-6

-4

-2

0

2

4

6

-4 -2 0 2 4

-6

-4

-2

0

2

4

6

-4 -2 0 2 4

6

6

33

>The foci are at the point (0, 0 27)

-6

-4

-2

0

2

4

6

-4 -2 0 2 4

Finding the Focus (foci)

>The foci lie on the major axis

a2 – b2 = c2

36 – 9 = c2

+ = 1 x2 y2

9 36

27 = c2

27 = c

>The distance to the foci from the center is c.

27

27

x2 y2

25 1 + = 1

-1

0

1

-5 -4 -3 -2 -1 0 1 2 3 4 5

Center = (0,0)Major Axis =10Minor Axis = 2

Foci: a2 – b2 = c2

25 – 1 = c2

24 = c2

24 or 2 6 = c

Foci:

(0 26, 0)

(x-2)2 (y+3)2

9 25 + = 1

Center = (2,-3)Major Axis =Minor Axis =

-10

-8

-6

-4

-2

0

2

4

-2 -1 0 1 2 3 4 5 65

5

33

-10

-8

-6

-4

-2

0

2

4

-2 -1 0 1 2 3 4 5 6

10

6

Foci: 25 – 9 = c2

16 = c2

4 = c

Foci are at (2, -3 4)

7x2 + 28x + 36y2 - 72y = 188

7x2 + 28x + 36y2 - 72y = 188

7(x2 + 4x ) + 36(y2 - 2y ) =188+ 4 + 1 +28+36

7(x + 2)2 + 36(y - 1)2 = 252252252

(x + 2)2 (y - 1)2

36 7+ = 1

(x+4)2 (y-1)2

36 7 + = 1Center =(-4, 1)Major Axis =12

Minor Axis = 2 7

-3

-1

1

3

5

-12 -10 -8 -6 -4 -2 0 2 4

7

7

66

-3

-1

1

3

5

-12 -10 -8 -6 -4 -2 0 2 4

Foci: a2 - b2 = c2

36 - 7 = c2

29 = c2

29 = c

Add c to the x of the center

(-4 29, 1)

-29 +29

-10

-8

-6

-4

-2

0

2

4

6

8

10

-6 -4 -2 0 2 4 6

Hyperbola x2 y2

9 36 - = 1

-10

-8

-6

-4

-2

0

2

4

6

8

10

-6 -4 -2 0 2 4 6

-10

-8

-6

-4

-2

0

2

4

6

8

10

-6 -4 -2 0 2 4 6

Center: (0,0)Out on the x-axis

Out on the y-axis

3 and -3

6 and -6

Draw asymptotes through corners of

boxslope = ± 6/3 or ± 2/1

Draw hyperbola on the x-axis not

crossing the dotted lines.

-10

-8

-6

-4

-2

0

2

4

6

8

10

-6 -4 -2 0 2 4 6

Foci: a2 + b2 = c2

x2 y2

9 36 - = 1

9 + 36 = c2

45 = c2

45 = c

(0 45, 0)

9y2 + 54y - 25x2 + 50 = 169

9y2 + 54y - 25x2 + 50 = 169

9(y2 + 6y ) - 25(x2 - 2x ) =169+ 9 + 1 +81-25

9(y + 3)2 - 25(x - 1)2 = 225225225

(y + 3)2 (x - 1)2

25 9- = 1

-15

-10

-5

0

5

10

-4 -2 0 2 4 6

-15

-10

-5

0

5

10

-4 -2 0 2 4 6

(y+3)2 (x-1)2

25 9 - = 1

Center : (1, -3)Slopes of Asymptotes : 5/3

Foci : a2 + b2 = c2

25 + 9 = c2

34 = c2

34 = c

Foci: (1,-3 34)