Equation of a Circle

This lesson covers the equation of a circle in the coordinate plane. Circles appear frequently in A-Level Mathematics, both as standalone problems and as part of larger questions involving tangents, intersections, and parametric curves. You must be able to work with both forms of the equation and convert between them fluently.

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Standard Form of the Equation of a Circle

A circle with centre $(a, b)$(a,b) and radius $r$r has equation

(x − a)² + (y − b)² = r²

This follows directly from the distance formula: a point (x, y) lies on the circle if and only if its distance from the centre (a, b) is exactly r.

Special case: If the centre is at the origin (0, 0), the equation simplifies to

x² + y² = r²

Example 1: Write down the equation of the circle with centre (3, −2) and radius 5.

(x − 3)² + (y + 2)² = 25

Example 2: State the centre and radius of the circle (x + 4)² + (y − 1)² = 49.

Centre = (−4, 1)    (note the signs carefully)
Radius = √49 = 7