Equation of a Circle: Standard and General Form

A circle is simply the set of every point in a plane that sits at one fixed distance, the radius $r$, from one fixed point, the centre $C(h,k)$. That single geometric idea translates directly into algebra: for any point $P(x,y)$ on the circle, the distance $CP$ must equal $r$, so

This is called the centre-radius (standard) form. If you expand the squares you get $x^2+y^2-2hx-2ky+(h^2+k^2-r^2)=0$. Renaming the constants as $g=-h$, $f=-k$ and $c=h^2+k^2-r^2$, every circle can be written in the tidier general form

and reading it backwards, any equation of this shape has centre $(-g,-f)$ and radius $r=\sqrt{g^2+f^2-c}$. Notice the radius only makes sense as a real length when $g^2+f^2-c\ge 0$ — if it equals zero the 'circle' shrinks to a single point, and if it is negative there is no real curve at all.

A natural question follows: given some general second-degree equation in $x$ and $y$, how do you know it is a circle and not, say, an ellipse or a pair of lines? The general second degree equation is $ax^2+2hxy+by^2+2gx+2fy+c=0$, and it represents a circle exactly when three conditions hold together: (i) the coefficients of $x^2$ and $y^2$ are equal and non-zero, $a=b\neq 0$; (ii) there is no $xy$ term, $h=0$; and (iii), after dividing through by $a$ so the equation is in the standard general form, $g^2+f^2-\dfrac{c}{a}\ge 0$ (writing $g,f,c$ for the normalised coefficients). This three-point checklist is worth keeping handy any time a problem asks you to identify or rule out a circle.

Worked Example. Does $2x^2+2y^2-8x+12y-2=0$ represent a circle? If so, find its centre and radius. …