General Equation of a Circle

Expanding the standard equation $(x-h)^2+(y-k)^2=r^2$ gives

which can be rewritten, after collecting terms, as

This is the general equation of a circle -- the equation of every circle can be written this way, and (subject to a condition below) every equation of this form represents a circle.

Recovering the centre and radius. Given a general equation $x^2+y^2+2gx+2fy+c=0$, the centre and radius are recovered by reversing the substitution above: since $g=-h$ and $f=-k$,

For the radius, rearrange $c=h^2+k^2-r^2=g^2+f^2-r^2$ to get $r^2=g^2+f^2-c$, so

Equivalently, this can be obtained directly by completing the square on $x^2+2gx$ and $y^2+2fy$: $(x+g)^2-g^2+(y+f)^2-f^2+c=0 \Rightarrow (x+g)^2+(y+f)^2=g^2+f^2-c$, matching the standard form with centre $(-g,-f)$ and $r^2=g^2+f^2-c$.

Condition for a real circle. Since $r^2$ must be positive for an actual circle to exist, the quantity $g^2+f^2-c$ must be positive:

• If $g^2+f^2-c>0$, the equation represents a genuine circle of radius $\sqrt{g^2+f^2-c}$.
• If $g^2+f^2-c=0$, the "radius" is $0$: the equation degenerates to the single point $(-g,-f)$ (matching the point degenerate conic of Section 1).
• If $g^2+f^2-c<0$, no real point satisfies the equation at all. …