Circle Properties

This lesson covers the key geometric properties of circles that are tested at A-Level. These properties connect algebra with geometry and are essential for solving problems involving tangents, chords, and intersections. You will use the equation of a circle together with these geometric facts to solve examination-style problems.

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Property 1: The Tangent is Perpendicular to the Radius

At any point P on a circle, the tangent to the circle at P is perpendicular to the radius drawn from the centre to P.

This is the most frequently used circle property at A-Level.

Application: To find the equation of the tangent at a point P on a circle:

1. Find the gradient of the radius from the centre to P.
2. The tangent gradient is the negative reciprocal.
3. Use y − y₁ = m(x − x₁) with point P.

Example 1: The circle C has equation (x − 2)² + (y − 3)² = 20. Find the equation of the tangent at the point P(6, 5).

Centre = (2, 3)

Gradient of radius CP = (5 − 3)/(6 − 2) = 2/4 = 1/2

Gradient of tangent = −2  (negative reciprocal)