Standard Equation

Circle with centre (h, k) and radius r. When centre is the origin: x² + y² = r².

A circle is the locus of all points in a plane at a fixed distance $r$ (the radius) from a fixed point $(h, k)$ (the centre).

Let $P(x, y)$ be any point on the circle. By the distance formula:

$$\sqrt{(x - h)^2 + (y - k)^2} = r$$

Squaring both sides:

$$(x - h)^2 + (y - k)^2 = r^2$$

This is the standard form. When the centre is at the origin $(0, 0)$:

$$x^2 + y^2 = r^2$$

The squaring step is valid because $r > 0$, so no extraneous solutions are introduced.