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: = = .
- Take the negative reciprocal.
Answer: Tangent gradient = −2
Higher Mathematics
Higher Maths Tangents to circles
Practise tangents to circles for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Use the perpendicular radius and tangent property to form tangent equations.
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
At a point of contact, the radius is perpendicular to the tangent
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.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.