Tangent to a circle

Definition

A tangent touches a circle at one point only and is perpendicular to the radius drawn to that point, so m_radius × m_tangent = −1.

Worked example

For x² + y² = 25 at (3; 4), the radius gradient is 4/3, so the tangent gradient is −3/4.

Tangent to x² + y² = 25 at (3; 4): the radius gradient is 4/3, so the tangent gradient is −3/4 and y − 4 = −¾(x − 3), which gives y = −¾x + 25/4.

Where it appears in the exam

Paper 2 asks for the equation of the tangent to a circle at a given point.

Step-by-step method

1. Find the centre of the circle.
2. Calculate the gradient of the radius to the point.
3. Use m_tangent = −1 ÷ m_radius.
4. Substitute the point into y − y₁ = m(x − x₁).

Common mistake

Using the radius's gradient as the tangent's gradient.

Related analytical geometry terms

• Angle of inclination

Frequently asked questions

What does tangent to a circle mean in Grade 12 Maths?

A tangent touches a circle at one point only and is perpendicular to the radius drawn to that point, so m_radius × m_tangent = −1.

Which NSC paper tests tangent to a circle?

Tangent to a circle belongs to the Analytical geometry section of CAPS Grade 12 Mathematics, which is examined in Paper 2.

How do I practise questions involving tangent to a circle?

Snap a question that uses tangent to a circle into the Snap&Learn AI solver, or work through the analytical geometry topic guide and NSC past papers.