Worked example 1
Find a circle tangent
A circle has centre (1, 2). Point P(5, 4) lies on the circle. Find the tangent gradient at P.
- Find the gradient of CP: 4 − 25 − 1 = 24 = 12.
- Take the negative reciprocal.
Answer: Tangent gradient = −2
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Higher Mathematics
Tangents to circles
Use radius and tangent gradients to form line equations at a point on a circle.
Before you start
- Find the centre and radius from a circle equation.
- Find a gradient from two points.
- Use perpendicular gradients in line equations.
Explanation
A tangent to a circle touches it at one point. The radius to that point is perpendicular to the tangent, so circle tangent questions are coordinate geometry questions with a perpendicular-gradient step.
The usual method is centre, radius gradient, tangent gradient, then line equation.
Visual support
Method and rules
- Radius gradient from centre C to point P.
- Tangent gradient = negative reciprocal of radius gradient
- Tangent through P: y − y₁ = m(x − x₁)
Worked examples
Worked example 2
Form the tangent equation
Use tangent gradient −2 through P(5, 4).
- Use y − b = m(x − a).
- Substitute m = −2 and P(5, 4)
So: y − 4 = −2(x − 5), so y = −2x + 14
Watch out
- Using the radius gradient as the tangent gradient.
- Finding the gradient from the wrong two points.
- Forgetting to use the point of contact in the line equation.
- Sign errors with the negative reciprocal.
Exam reminder
A diagram is often enough to spot whether your tangent gradient sign is plausible. The tangent must be perpendicular to the radius at the contact point.