Circles, Ellipses, and Hyperbolas

Circles – Standard equation is (x-h)2 + (y – k)2 = r2 where (h,k) is the center of the circle and r is the length of the radius.

Eg x2 + y2 = 16 is a circle whose center is at (0,0) and radius is 4.

EG (x + 5)2 + (y - 3)2 = 12 is a circle whose center is at (-5, 3) and radius of .

Sometimes you have to complete the square to find the center of the circle.

EG X2 + y2 + 6x – 8y = 4

X2 + 6x + y2 – 8y = 4

X2 + 6x + 9 + y2 – 8y + 16 = 4 + 9 + 16

(x +3)2 + (y – 4)2 = 29 is a circle whose center is at (-3,4) and radius of

EG x2 + y2 -8x = 8

X2 -8x + 16 + y2 = 8 + 16

(x – 4)2 + y2 = 24 is a circle whose center is at (4,0) and radius of

Ellipses – Standard equation is (x-h)2/a2 + (y-k)2/b2 = 1 where (h,k) is the center and the vertices are at

(h+a,k), (h-a,k), (h,k+b), (h,k-b).

EG x2/4 + y2/9 = 1 is an ellipse whose center is at (0,0) and vertices at (2,0), (-2,0), (0,3), (0,-3)

EG (x+2)2/9 + (y – 1)2/25 = 1 is an ellipse whose center is at (-2,1) and vertices are (1,1), (-5,1), (-2,6), (-2,-4)

EG 5x2 + 20y2 = 20

5x2 /20 + 20y2/20 = 20/20

X2/4 + y2 = 1 is an ellipse whose center is at (0,0) and vertices are (2,0), (-2,0), (0,1), (0,-1)

EG 16x2 + 8y2 = 32

16x2/32 + 8y2/32 =32/32

(-,0), (0,2), (0,-2)

5x2 +10x + 2y2 – 8y = 40 5(x2 + 2x + 1) +2(y2 – 4y + 4) = 40 + 5 +8

(x + 1)2/10.6 + (y – 2)2/26.5 = 1 is an ellipse whose center is at (-1, 2) and vertices are

(-1 + , 2), (-1 – , 2), (-1, 2 + ), (-1, 2 – )

Hyperbolas have a standard equation of (x-h)2/a2 – (y-k)2/b2 = 1 with center at (h,k) vertices at(h+a,k) and (h-a,k) and asymptotes( y – k) = b/a(x – h) and( y – k) = -b/a( x – h) that open to the right and left OR

-(x-h)2/a2 + (y-k)2/b2 = 1 with center at (h,k) vertices at(h,k+b) and (h,k-b) and asymptotes (y – k) =a/b (x – h) and (y – k) = -a/b( x – h) that opens up or down

EG x2/4 – y2/9 = 1

Center at (0,0) vertices (2,0) and (-2,0) asymptotes y = 3/2 x and y = -3/2 x

EG - x 2 + y2/16 = 1

Center is at (0,0) vertices at (0,4) and (0, -4) asymptotes y = ¼ x and y = - ¼ x

EG (x – 2)2/9 – (y + 3)2 = 1 is a hyperbola that opens right and left. Center is at (2, -3) vertices at (5,-3) and (-1,-3) and asymptotes y + 3 = 1/3(x-2) and y + 3 = - 1/3(x-2)

EG –(x + 5)2/5 + (y- 2)2/4 = 1 is a hyperbola that opens up and down. Center is at (-5, 2) vertices at (-5, 4) and (-5, 0) and asymptotes y – 2 = /2(x + 5) and y – 2 = -/2(x + 5)