What is important to know about Equation of a circle?

x² + y² = r² describes a circle centred at (0, 0) with radius r.

What is important to know about Equation of a circle?

The radius at a point is perpendicular to the tangent at that point.

What is important to know about Equation of a circle?

Gradient of the radius from (0,0) to (a,b) is b/a.

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Algebra · Higher

Equation of a circle

A circle centred at the origin with radius r has equation x² + y² = r². At Higher GCSE you must also find the equation of a tangent to a circle at a given point.

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Revising equation of a circle the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Equation of a circle GCSE” on the site for the complete guide.

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Key facts to remember

1x² + y² = r² describes a circle centred at (0, 0) with radius r.
2The radius at a point is perpendicular to the tangent at that point.
3Gradient of the radius from (0,0) to (a,b) is b/a.
4Gradient of the tangent is the negative reciprocal: −a/b.

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Formulas

Circle centred at origin

x² + y² = r²

Tangent gradient at (a,b)

m_tan = −a/b

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Worked examples

Example 1

Find the equation of the tangent to x² + y² = 25 at the point (3, 4).

Working

Radius gradient: 4/3.
Tangent gradient: −3/4.
Use y − 4 = −3/4 (x − 3).
Expand: y − 4 = −3/4 x + 9/4.
y = −3/4 x + 25/4.

Answery = −3/4 x + 25/4

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Common mistakes

✗Using the radius gradient for the tangent instead of the negative reciprocal.

✗Forgetting to square r in x² + y² = r².

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Exam tips

✓Work out the radius gradient first, then flip and change sign for the tangent.

✓Check: the tangent should pass through the given point.

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