The Equation of a Circle

A circle is every point sitting a fixed distance from the centre. Writing that with the distance formula and squaring both sides gives:

(x − h)² + (y − k)² = r²

with centre (h, k) and radius r.

Two things to watch. The signs flip: (x − 3)² means h = 3, and (y + 2)² means k = −2. And the right side is r², so a 25 there means the radius is 5, not 25.

If the equation is expanded instead — something like x² + y² − 6x + 4y − 12 = 0 — complete the square on the x terms and the y terms separately to get back to the useful form.

To test whether a point is inside, on, or outside, substitute it: less than r² is inside, equal is on, greater is outside.

The equation follows directly from the distance formula: the locus of points at distance r from (h,k) satisfies √((x−h)² + (y−k)²) = r, and squaring gives the standard form (x−h)² + (y−k)² = r². It is the Pythagorean theorem stated as a locus.

The general form x² + y² + Dx + Ey + F = 0 converts to standard form by completing the square in both variables. The result (x + D/2)² + (y + E/2)² = (D² + E² − 4F)/4 has a real circle only when the right side is positive; zero gives a single point and negative gives no real graph.

Substituting a point classifies its position: the left-hand side evaluates to less than, equal to, or greater than r² for interior, boundary and exterior points respectively.

The circle is the conic section with equal coefficients on x² and y² and no xy term, and is the special case of an ellipse with equal axes. Combining it with a line and solving the resulting system determines whether the line is a secant (two solutions), tangent (one) or external (none) — an algebraic version of the tangent condition.