GCSEMathsAQARevision Notes2. AlgebraEquation of a CircleEquation of a Tangent

Equation of a Tangent (AQA GCSE Maths)

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Daniel I

Last updated

16 March 2023

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Equation of a Tangent

How do we find the equation of a tangent to a circle?

First, make sure you are familiar with Equations of Straight Lines and Perpendicular Gradients
A TANGENT just touches a circle (but does not cross it)
The gradient of the tangent at point P is PERPENDICULAR to the gradient of the radius OP
- remember, perpendicular gradients multiply to -1 (they are negative reciprocals)
So if P is a point $(x, y)$ on the circumference, then the gradient of the radius OP is $\frac{y - 0}{x - 0} = \frac{y}{x}$ .
- Therefore the gradient of the tangent to the circle at P is $- \frac{x}{y}$
From here, use $y = m x + c$ to find the equation of the tangent

Examiner Tip

Solving simultaneous equations of circle and tangent only gives ONE SOLUTION
- so if you are asked to show a line is a tangent, solve the simultaneous equations and show there is only one solution
ALWAYS draw a diagram to help!

Worked example

Circle Radius Tangent (before) Circles - Finding Tangents WE 1

Circles - Finding Tangents WE 2a

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Author: Daniel I

Expertise: Maths

Daniel has taught maths for over 10 years in a variety of settings, covering GCSE, IGCSE, A-level and IB. The more he taught maths, the more he appreciated its beauty. He loves breaking tricky topics down into a way they can be easily understood by students, and creating resources that help to do this.