Circle Ellipse HyperbolaParabola Conic Sections

Circles x 2 + y 2 = 16 center: (0,0) radius: Ex. 1 Standard form: (x – h) 2 + (y – k) 2 = r 2 center: (h, k) radius: r (x – 0) 2 + (y – 0) 2 = 4 2

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

x 2 + y 2 = 60 center: (0,0) radius:

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

Find the center and radius of the circle: x 2 + y x – 6y + 33 = 0 x x + y 2 – 6y = -33 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 + 8x – 2y + 13 = 0 x 2 + 8x + y 2 – 2y = -13 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 Center = (0,0) Major Axis = 10 Minor Axis = 4 x 2 y = Standard form: + = 1 (x - h) 2 (y - k) 2 a 2 b 2

Ellipse Center = (0,0) Major Axis = 10 Minor Axis = 4 x 2 y = Standard form: + = 1 (x - h) 2 (y - k) 2 a 2 b 2 Foci will lie on the major axis c units from the center a 2 – b 2 = c 2 25 – 4 = c 2 21 = c 2  21 = c Foci: (0  21, 0) -  21  21

36x 2 +9y 2 – 72x +54y = 207 Center =(1, -3) Major Axis =12 Minor Axis = (x 2 – 2x ) + 9(y 2 + 6y ) = x 2 – 72x + 9y 2 +54y = (x – 1) 2 + 9(y + 3) 2 = (x – 1) 2 (y + 3) = 1

>The foci are at the point (1, -3   27) Finding the Focus (foci) >The foci lie on the major axis a 2 – b 2 = c 2 36 – 9 = c 2 + = 1 (x-1) 2 (y+3) = c 2  27 = c  27 >The distance to the foci from the center is c.

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

(x-2) 2 (y+3) = 1 Center =(2,-3) Major Axis = Minor Axis =

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

Moving the center >The center will be at (1, -3) >The major axis will be 12 Units long and be parallel to The y-axis. >The minor axis will be 6 units long and be parallel to the x-axis = 1 (x-1) 2 (y+3) = 1 (x-1) 2 (y+3) = 1 (x-1) 2 (y+3)

7x x + 36y y = 104 7(x 2 + 8x ) + 36(y 2 - 2y ) = (x + 4) (y - 1) 2 = (x + 4) 2 (y - 1) = 1

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

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

Foci: a 2 + b 2 = c 2 x 2 y = = c 2 45 = c 2  45 = c (0  45, 0)

9y y - 25x = 169 9(y 2 + 6y ) - 25(x 2 - 2x ) = (y + 3) (x - 1) 2 = (y + 3) 2 (x - 1) = 1

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