MODERATE EXERCISES

(3) Describe the solution curve represented by each of the following equations and sketch the graph:

 (a) x2 + y2 - 4x - 6y + 13 = 0

 (b) y2 + 4y + 2 = x2 + 2x

 (c) y2 - 5y + 3 = 2x + (y - 3)2

 (a) x2 + y2 - 4x - 6y + 13 = 0
     (x2 - 4x) + (y2 - 6y = -13
     (x - 2)2 - 4 + (y - 3)2 - 9 = -13
     (x - 2)2 + (y - 3)2 = 0 ==> x = 2 , y = 3
     
     This is a degenerate circle

     


 (b) y2 + 4y + 2 = x2 + 2x
     (y + 2)2 + 2 - 4 = (x + 1)2 - 1
     (y + 2)2 - (x + 1)2 = 1

     This is a hyperbola.

     


 (c) y2 - 5y + 3 = 2x + (y - 3)2
     y2 - 5y + 3 - (y - 3)2 = 2x

     2x = y2 - 5y + 3 - (y2 - 6x + 9)
     2x = y - 6

      y = 2x + 6

     This is a line.