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The equation of a circle, (x − a)² + (y − b)² = r², gives its centre and radius. This topic covers completing the square to find them and the circle properties used in coordinate problems. It appears across A-Level and iAL Pure Maths with the major exam boards. Learn it free on eClassroom through guided worked examples and instantly-marked exam-style questions.
How you do it
The equation is x minus a all squared plus y minus b all squared equals r squared, with centre a, b and radius r. If it is given expanded, complete the square in x and in y to recover the centre and radius. The signs flip, so a plus inside the bracket means a negative coordinate.
Before you start
Everything here rests on completing the square and distance. It belongs to the Pure 2 specification. Expect to use it again in tangents, chords and intersections. It is part of the Coordinate Geometry strand.
What you learn
A circle is the set of points a fixed distance from a fixed centre — and that single sentence is its equation. To recover them, complete the square in x and in y separately — then read them off as before. Exam questions sometimes demand the answer in the expanded form x²+y²+2fx+2gy+c=0 with integer coefficients — so be ready to multiply out and clear fractions. If BC is a diameter and A is any other point on the circle, then angle BAC=90°. The course works through the equation of a circle, completing the square, A circle from a diameter, the angle in a semicircle and problem solving with circles. Every section has instant-feedback practice questions matched to Edexcel International A-Level and 9MA0 Pure, and the course finishes with full exam-style questions you mark yourself.
Related Topics
Circles — Equation & Properties is a Coordinate Geometry topic in the A-level syllabus, covered here at the AS year. The course is built from 5 concept sections, 5 worked examples and 5 check points before the exam-style set. It sits alongside Circles — Tangents & Chords, Distance & Midpoint, Parametric Equations and Straight Lines in the same strand. Endless extra practice on this strand is available on the Coordinate Geometry revision engine, which generates a fresh question each time.
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