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Equation of a tangent

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The equation of a tangent is crucial in understanding how straight lines can "just touch" curves at a single point. In real life, this concept helps in designing roads, bridges, and even roller coasters, where smooth transitions between curved and straight sections are essential for safety and comfort. Jump to the questions

Practise now

Numerical values should be entered as integers or as proper or improper fractions (not decimals nor mixed numbers).


Topic guide

What this worksheet practises

This worksheet provides practice on finding the equation of a tangent to a circle. A tangent is a straight line that touches the outside of a circle at exactly one point. This requires combining circle theorems (specifically that the tangent meets the radius at 90°) with perpendicular line geometry.

Key method

You must find the gradient of the radius first, and then use the negative reciprocal rule to find the gradient of the tangent.

  • Find the centre of the circle (usually the origin (0,0) at GCSE level) and the coordinates of the point where the tangent touches.
  • Calculate the gradient of the radius connecting the centre to that point (change in y ÷ change in x).
  • Find the negative reciprocal of the radius gradient. This is the gradient of your tangent line.
  • Write your tangent equation as y = mx + c.
  • Substitute the coordinate of the touching point into the equation to calculate 'c'.

Worked example

A circle has equation x² + y² = 25. Find the equation of the tangent at the point (3, 4).

Step 1: The circle is centred at (0, 0). Calculate the gradient of the radius from (0, 0) to (3, 4).

Gradient of radius = 4 / 3.

Step 2: Find the tangent gradient. It is perpendicular, so use the negative reciprocal.

Tangent gradient (m) = −3/4. So, y = −3/4 x + c.

Step 3: Substitute the point (3, 4) to find 'c'.

4 = −3/4(3) + c

4 = −9/4 + c

c = 4 + 9/4 = 16/4 + 9/4 = 25/4.

Step 4: Write the final equation.

y = −3/4 x + 25/4 (or 4y = −3x + 25).

Common mistakes to avoid

The most fatal error is using the gradient of the radius as the gradient of the tangent. Remember the circle theorem: the tangent is always perpendicular to the radius at the point of contact. You must flip the fraction and change the sign.

Things to remember

At standard GCSE level, the circle is almost always centred at the origin (0,0). If the circle equation is x² + y² = r², the gradient of the radius to point (x, y) is simply y/x.