### Equation of a circle

#### Circles with centre the origin

Consider a circle in the coordinate plane with centre the origin and radius r.

If P(x, y) is a point on the circle, its distance from the origin is r and, by Pythagoras' theorem, \(x^2 + y^2 = r^2\).

Conversely, if a point P(x, y) in the plane satisfies the equation \(x^2 + y^2 = r^2\), its distance from O(0, 0) is r, so it lies on a circle with centre the origin and radius r.

#### Summary

The circle with centre O(0, 0) and radius r has equation \(x^2 + y^2 = r^2\).