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# Equation of a tangent
- URL: https://www.esheets.io/equation-of-a-tangent/
- Published: 2024-09-25T23:00:11.000Z
- Updated: 2026-06-21T16:57:30.000Z
- Author: Richard Linnington
- Tags: Maths, Algebra

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)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise the equation of a tangent with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Using the radius to the point of contact.
- Recognising that the tangent is perpendicular to the radius.
- Using the negative reciprocal gradient.
- Finding the equation of the tangent line.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

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.