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Equation of a Circle (OCR GCSE Maths) : Revision Note

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Equation of a Circle

What is the equation of a circle?

  • A circle centered on the origin with radius r has the equation x2 + y2 = r2

  • (a, b) lies on the circle if a2 + b2 = r2

    • If a2 + b2 < r2 then (a, b) lies inside the circle

    • If a2 + b2 > r2 then (a, b) lies outside the circle

  • The circle cuts the x- and y-axes at ±r (“plus or minus r”)

  • The diameter = 2r and the circumference = 2πr

Examiner Tips and Tricks

  • if asked for the radius, don't forget to square root r2

    • e.g. for x2 + y2 = 10, the radius is √10, not "10"

Worked Example

The diagram shows a circle, centred on the origin.

Circles, IGCSE & GCSE Maths revision notes

(a) Write down the equation of the circle.

Identify the radius by checking where the circle crosses the coordinate axes

r equals 5

Substitute this into x squared plus y squared equals r squared

x squared plus y squared equals 5 squared

x squared plus y squared equals 25

(b) Does the point P open parentheses 3 comma space 4 close parentheses lie on, inside or outside the circle? You must show working to support your answer.

Substitute x equals 3 and y equals 4 into x squared plus y squared

table row cell x squared plus y squared end cell equals cell 3 squared plus 4 squared end cell row blank equals cell 9 plus 16 end cell row blank equals 25 end table

Since 3 squared plus 4 squared equals 25, the point P open parentheses 3 comma space 4 close parentheses lies on the circle

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    Amber

    Author: Amber

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    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.