Equation of a tangent – GCSE Maths

Higher · Grade 8 · Non-calculator

Jump to the questions and answers · Download the worksheet (PDF)

For the circle $x^2+y^2=r^2$, the radius to $(x_1,y_1)$ on the circle has gradient $\frac{y_1}{x_1}$, so the tangent is perpendicular: $m=-\frac{x_1}{y_1}$. Then $y-y_1=m(x-x_1)$, or $x{x}_1+y{y}_1=r^2$ as a shortcut for this circle. For $(x-a{)}^2+(y-b{)}^2=r^2$, the radius gradient is $\frac{y_1-b}{x_1-a}$.

Confirm the point sits on the circle before you start. Rearrange to $y=mx+c$ if asked. A second tangent from an external point is a later, harder variant — at this grade you are usually given the point of contact.

For $x^2+y^2=r^2$ at $(x_1,y_1)$, the radius has gradient $\frac{y_1}{x_1}$, so the tangent has gradient $-\frac{x_1}{y_1}$. Then $y-y_1=m(x-x_1)$, or the neat form $x{x}_1+y{y}_1=r^2$. First check the point sits on the circle.

Objectives

• Find the gradient of the radius.
• Flip it to the tangent gradient.
• Write y - y1 = m(x - x1), then tidy it.

Key points

• Recognise $x^2+y^2=r^2$ as a circle with centre $(0,0)$ and radius $r$.
• Check that a given point lies on the circle by substituting.
• Find the gradient of the radius to the point, then take the negative reciprocal for the tangent.
• Write the tangent equation using $y-y_1=m(x-x_1)$ and tidy it into the requested form.

Worked examples

Questions and answers

GCSE Maths equation of a tangent questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Using the radius gradient as the tangent gradient

   The tangent is perpendicular to the radius, so use the negative reciprocal

2. Flipping the gradient but keeping the same sign

   The product of perpendicular gradients is -1, so change the sign as well

3. Leaving the equation as y - y1 = m(x - x1) when y = mx + c is asked

   Expand the bracket and collect the constant, keeping exact fractions

Exam tips

• Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Write 'tangent is perpendicular to radius, so gradient $=-\frac{a}{b}$' — this sentence is usually a method mark by itself.
• Show the substitution check that the point lies on the circle; it takes one line and protects against a misread coordinate.

Printable worksheet

The same questions are in a PDF you can print. Answers start on a later page.

Download worksheet (PDF)

Or use Print in your browser. Answers under each question are included when you print.

Questions students ask

Related topics

• Circle theorem proofs
  Grade 8 · Higher
• Vector proof questions
  Grade 8 · Higher
• 3D Pythagoras & trigonometry
  Grade 7 · Higher
• Area of any triangle
  Grade 7 · Higher
• Combined transformations
  Grade 7 · Higher

More on Equation of a tangent

• Equation of a tangent worked examples and method
  The method in order, key facts and the full lesson explanation
• Equation of a tangent practice questions
  A second set of questions with answers, plus quick checks

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.