Under what condition on the coefficients a, b, and c does the equation x2 + y2 + ax + by + c = 0 represent a circle?

Question

When the condition is satisfied, find the radius of the circle. When the condition is satisfied, find the center of the circle.

Answer 1

x^2 + ax + y^2 +by + c= 0

subtract c from both sides

x^2 +ax +y^2 +by = -c

add half a squared and half b squared to both sides

x^2 + ax + (a/2)^2 + y^2 + by + (b/2)^2 = -c+ a^2/4 + b^2/4

it's a circle when a, b and c have the following relationship

(x+a/2)^2 + (y+b/2)^2 = a^2/4 + b^2/4 - c

center of the circle is (-a^2/2, - b^2/2)

radius = sqr(a^2/4 +b^2/4 -c)

with a^2/4 +b^2/4-c > 0

a^2 +b^2 >4c