General Equation of a Circle

A circle of centre $(h_0,k_0)$ and radius $r$ satisfies, by the plain distance definition of a circle (every point on it is at distance $r$ from the centre):

Expanding this and collecting terms always produces an equation of the form

with no $xy$ term and equal (here, unit) coefficients on $x^2$ and $y^2$ — this is the general equation of a circle. Comparing the expanded form with this general form gives:

(the equation represents a real circle only when $g^2+f^2-c > 0$; if it equals zero the "circle" is a single point, and if it is negative there is no real curve at all).

Given any point $(x_1,y_1)$, substituting it into the left side of the general equation, $S_1 = x_1^2+y_1^2+2gx_1+2fy_1+c$, tells us where the point lies relative to the circle: $S_1 < 0$ means inside, $S_1 = 0$ means on the circle, and $S_1 > 0$ means outside — a very quick test once a circle's equation is known. …