Rukasha Maths

GCSE maths Beta

Circle tangents and graph translations · Higher notes

Higher · Algebra · AQA references A9, A13, A16

The equation x² + y² = r² describes points a distance r from the origin, using Pythagoras theorem. A tangent touches a circle at one point and is perpendicular to the radius there. Find the radius gradient, use the perpendicular gradient, then substitute the contact point to locate the line. Graph translations use another useful connection: replacing x with x - a shifts a graph a units right, while adding b to the output shifts it b units up.

Learning objectives

  • Find a tangent to a circle using a perpendicular radius.
  • Interpret horizontal and vertical graph translations.

Key ideas

A circle centred at the origin has radius given by the square root.

x2+y2=r2x^2+y^2=r^2

For two non-vertical perpendicular lines, gradients are negative reciprocals.

m1m2=−1m_1m_2=-1

A horizontal radius has a vertical tangent; a vertical radius has a horizontal tangent.

The inner subtraction gives a horizontal shift; the outer addition gives a vertical shift.

y=f(x−a)+bshifts by(ab)y=f(x-a)+b\quad\text{shifts by}\quad\binom ab

Worked examples

Example 1

Find the tangent to the circle at the given point.

x2+y2=25,P=(3,4)x^2+y^2=25,\qquad P=(3,4)
32+42=25,mOP=43,mtangent=−343^2+4^2=25,\qquad m_{OP}=\frac43,\quad m_{\text{tangent}}=-\frac34
y−4=−34(x−3)y-4=-\frac34(x-3)

Answer

3x+4y=253x+4y=25

Example 2

Find the turning point of the translated quadratic.

y=(x+2)2−3,x+2=0⇒x=−2y=(x+2)^2-3,\qquad x+2=0\Rightarrow x=-2

Answer

(−2,−3)(-2,-3)

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Find a tangent equation at a point on a circle or translate a graph.

AO2 · Reasoning

Use perpendicular gradients to justify a tangent, or explain a translation.

AO3 · Problem solving

Combine circle and line equations to determine a geometric relationship.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Horizontal and vertical lines need their special tangent cases. Read the horizontal translation with the opposite sign inside the bracket.

Try it yourself

Attempt each question before opening its answer.

Check 1: What is the radius of x² + y² = 81?

Answer: 9

The right-hand side is r², so take its square root.

Check 2: What is the turning point of y = (x + 2)² - 3?

Answer: (-2, -3)

The square is zero at x = -2; the vertical shift is -3.

Explain it in your own words

  • Explain why the radius is the square root of the constant in a circle equation.
  • Write the steps for finding a tangent at a non-axis point.
  • Sketch how y = (x - 1)² + 2 moves from y = x².

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes

Specification source

AQA Algebra specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.