Edexcel · GCSE Maths · 1MA1 · Higher only

M21 · Circle equations and tangents

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Revision notes, worked examples and methods for circle equations and tangents.

Notes and quizzes ready · 50 questions · Revision video ready.

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Higher — equations of circles

A circle with centre at the origin and radius r has equation x² + y² = r². It follows from Pythagoras applied to the horizontal and vertical coordinates.
Radius perpendicular to a tangent at three four
Enlarge diagram: Radius perpendicular to a tangent at three four
Worked example: x² + y² = 25 has centre (0, 0) and radius 5, not 25.
A point is on the circle when its coordinates satisfy the equation. (3, 4) is on x² + y² = 25 because 9 + 16 = 25.
If x² + y² is smaller than r², the point lies inside the circle; if larger, it lies outside.
Given x on the circle, y = ±√(r² − x²), where |x| ≤ r. Most vertical lines through the circle meet it twice.

Higher — tangents

A tangent touches a circle at one point and is perpendicular to the radius there. Find the radius's gradient, then take the negative reciprocal for the tangent.
Worked example: At (3, 4) on the radius-5 circle, the radius gradient is 43. The tangent gradient is −34.
Using y = mx + c, 4 = −34 × 3 + c gives c = 254. The tangent is y = −34x + 254, or 3x + 4y = 25.
At (r, 0), the radius is horizontal and the tangent is vertical: x = r. At (0, r), the tangent is horizontal: y = r. Avoid attempting a reciprocal of zero.
To check a tangent equation, confirm it passes through the stated point and its gradient is perpendicular to the radius.

Higher — intersections

Substitute a line equation into the circle equation to find intersections. A tangent produces a repeated root, representing a single contact point.
Worked example: For y = 3 on x² + y² = 25, x² + 9 = 25 gives x = ±4. The two points are (−4, 3) and (4, 3).
Circle equations describe a full locus, not one function y of x unless a branch is selected. Keep both branches when the question concerns the whole circle.

Test yourself

50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.

Watch M21 · Circle equations and tangents

Revise circle equations and tangents with this narrated video. Use the player controls to pause, seek, adjust the volume or mute. Turn English captions on or off using the captions menu.