Conic Sections Circle Hyperbola Ellipse Parabola 2 Circles

Conic Sections Circle Hyperbola Ellipse Parabola

2 Circles 2 x + y = 16 center: (0, 0) radius: 4 4 4

2 2 x + y = 36 center: (0, 0) radius: 6 6

2 2 x + y = 60 center: (0, 0) radius: 60 2 15 7. 746

2 2 (x-5) + (y+2) = 9 center: (5, -2) radius: 3 3

Find the center and radius of the circle: x 2 + y 2 + 10 x – 6 y + 33 = 0 x 2 + 10 x + 25 + y 2 – 6 y + 9 = -33 + 25 + 9 Fill in the blanks by completing the square If you add 25 and 9 to one side, you must add 25 and 9 to the other side 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: x 2 + y 2 + 8 x – 2 y + 13 = 0 x 2 + 8 x + 16 + y 2 – 2 y + 1 = -13 + 16 + 1 Fill in the blanks by completing the square If you add 16 and 1 to one side, you must add 16 and 1 to the other side Factor the left hand side (x + 4)2 + (y – 1)2 = 4 Center: (-4, 1) Radius: 2

Ellipse x 2 y 2 + =1 25 4 5 Center = (0, 0) Major Axis = 10 2 Minor Axis = 4 5 2

x 2 y 2 + =1 9 36 Center = (0, 0) Major Axis =12 Minor Axis = 6 6 3 3 6

Finding the Focus (foci) x 2 + y 2 = 1 >The foci lie on the major axis 9 36 >The distance to the foci from the center is c. 27 a 2 – b 2 = c 2 36 – 9 = c 2 27 = c >The foci are at the point (0, 0 27)

x 2 y 2 + =1 25 1 Foci: a 2 – b 2 = c 2 25 – 1 = c 2 24 or 2 6 = c Center = (0, 0) Major Axis =10 Minor Axis = 2 Foci: (0 2 6, 0)

(x-2)2 (y+3)2 + =1 9 25 3 5 5 3 Center = (2, -3) Major Axis = 10 Minor Axis = 6 Foci: 25 – 9 = c 2 16 = c 2 4=c Foci are at (2, -3 4)

7 x 2 + 28 x + 36 y 2 - 72 y = 188 7(x 2 + 4 x + 4) + 36(y 2 - 2 y + 1 ) =188 +28+36 7(x + 2)2 + 36(y - 1)2 = 252 252 (x + 2)2 + (y - 1)2 = 1 36 7

(x+4)2 36 + (y-1)2 7 6 - 29 Center = (-4, 1) Major Axis =12 Minor Axis = 2 7 =1 6 + 29 7 7 Foci: a 2 - b 2 = c 2 36 - 7 = c 2 29 = c 2 29 = c Add c to the x of the center (-4 29, 1)

Hyperbola x 2 y 2 =1 9 36 Center: (0, 0) Out 3 and -3 on the x-axis Out 6 and -6 on the y-axis Draw asymptotes through corners of box slope = ± 6/3 or ± 2/1 Draw hyperbola on the x-axis not crossing the dotted lines.

x 2 y 2 =1 9 36 Foci: a 2 + b 2 = c 2 9 + 36 = c 2 45 = c 2 45 = c (0 45, 0)

9 y 2 + 54 y - 25 x 2 + 50 = 169 9(y 2 + 6 y + 9 ) - 25(x 2 - 2 x + 1) =169 +81 -25 9(y + 3)2 - 25(x - 1)2 = 225 225 (y + 3)2 (x - 1)2 =1 25 9

(y+3)2 (x-1)2 =1 25 9 Center : (1, -3) Slopes of Asymptotes : 5/3 Foci : a 2 + b 2 = c 2 25 + 9 = c 2 34 = c 2 34 = c Foci: (1, -3 34)