M21 · Circle equations and tangents | Edexcel GCSE Maths

Revise the key ideas

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.

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.

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.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.