INTRODUCTORY EXERCISES

(1) By completing the square, show that the 
following equations represent a circle and find its radius and center.

 (a) x2 + y2 + 2x - 6y + 7 = 0

 (b) x2 + y2 - 4x + 10y + 13 = 0

 (c) 16x2 + 16y2 + 8x + 32y + 1 = 0

 (a) x2 + y2 + 2x - 6y + 7 = 0

     (x2 + 2x) + (y2 - 6y) = -7

     (x + 1)2 - 1 + (y - 3)2 - 9 = -7

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

     center = (-1,3)  radius = 


 (b) x2 + y2 - 4x + 10y + 13 = 0

     (x2 - 4x) + (y2 + 10y) = -13

     (x - 2)2 - 4 + (y + 5)2 - 25 = -13

     (x - 2)2 + (y + 5)2 = 16

     center = (2,-5)  radius = 4


 (c) 16x2 + 16y2 + 8x + 32y + 1 = 0

     16[(x2 + 1/2x) + (y2 + 2y)] = -1

     16[(x + 1/4)2 + (y + 1)2 -1/16 - 1] = -1

     (x + 1/4)2 + (y + 1)2 = -1/16 + 1/16 + 1

     (x + 1/4)2 + (y + 1)2 = 1

     certer = (-1/4,-1)  radius = 1