Does every equation of the form x^2+mx+y^2+ny=p represent a circle? Explain

Question

Please help me find the answer please.

Answer 1

The answer is NO.  For example, let m = 2, n = 2 and p = -2.

 

Then, we have x²+2x+y²+2y = -2

 

        Using the completing the square technique, we have:

 

              x²+2x+1+y²+2y+1=0

 

              (x+1)²+(y+1)²=0   The point (-1,-1) satisfies this equation,                 but for any other point, the left side of the equation is                         positive.  So, the graph of the equation is the single point                   (-1,-1), not a circle.