Equation of a Circle — CAIE A-Level Pure Mathematics 1 (9709) — Mathematics

Introduction

The equation of a circle is a core topic in the Coordinate Geometry unit of Pure Mathematics 1. It connects the geometric definition of a circle (all points at a fixed distance from a centre) directly to algebra. In 9709 examinations, questions on circles appear regularly and often combine with straight-line work — for example, finding tangents, normals, or points of intersection. Mastering both standard forms and the ability to switch between them is essential for full marks.

Core Concept

The Geometric Definition

A circle is the locus of all points that are a fixed distance $r$ (the radius) from a fixed point $(a, b)$ (the centre).

If a general point $(x, y)$ lies on the circle, then by the distance formula:

\sqrt{(x-a)^2 + (y-b)^2} = r

Squaring both sides gives the standard equation.

Form 1 — Centre–Radius Form

\boxed{(x - a)^2 + (y - b)^2 = r^2}

This form immediately reveals the centre $(a, b)$ and radius $r$ .

Sign convention: The centre coordinates appear with the opposite sign inside the brackets. The centre of $(x-3)^2 + (y+2)^2 = 25$ is $(3, -2)$ , not $(-3, 2)$ .

Form 2 — Expanded (General) Form

Expanding Form 1 and rearranging yields the general form:

\boxed{x^2 + y^2 + 2gx + 2fy + c = 0}

Here the centre is $(-g, -f)$ and the radius is $r = \sqrt{g^2 + f^2 - c}$ , provided $g^2 + f^2 - c > 0$ .

The two forms are entirely equivalent; the choice of which to use depends on what information is given or required.

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