GCSE — mathsHigher2 min read

Equation of a Circle

Understand the equation of a circle centred at the origin for GCSE Maths. Apply x² + y² = r² and find tangent equations.

Tutor AI Team2026-02-25T01:46:49.374Z

The equation $x^2 + y^2 = r^2$ represents a circle centred at the origin with radius $r$ . This is a Higher-tier GCSE topic that links coordinate geometry with circle properties.

Core Concepts

Standard Equation

$x^2 + y^2 = r^2$

Centre: $(0, 0)$ . Radius: $r$ .

Points on the Circle

A point $(a, b)$ lies on $x^2 + y^2 = 25$ if $a^2 + b^2 = 25$ .

$(3, 4)$ : $9 + 16 = 25$ ✓. $(2, 5)$ : $4 + 25 = 29$ ✗.

Tangent to a Circle

A tangent at point $P$ is perpendicular to the radius at $P$ .

Finding the tangent at $(3, 4)$ on $x^2 + y^2 = 25$ :

Gradient of radius $OP = \frac{4}{3}$ .
Gradient of tangent = $-\frac{3}{4}$ (perpendicular).
Equation: $y - 4 = -\frac{3}{4}(x - 3)$ .

Worked Example: Example 1

Problem

Does $(5, 12)$ lie on $x^2 + y^2 = 169$ ? $25 + 144 = 169$ ✓.

Solution

Worked Example: Example 2

Problem

Find the equation of the tangent to $x^2 + y^2 = 20$ at $(2, 4)$ .

Radius gradient: $\frac{4}{2} = 2$ . Tangent gradient: $-\frac{1}{2}$ .

$y - 4 = -\frac{1}{2}(x - 2)$ → $y = -\frac{1}{2}x + 5$ .

Solution

Practice Problems

1. Write the equation of a circle with centre $(0,0)$ and radius 7.
1. Does $(1, \sqrt{8})$ lie on $x^2 + y^2 = 9$ ?
1. Find the tangent at $(6, 8)$ on $x^2 + y^2 = 100$ .

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Key Takeaways

✓
Circle centred at origin: $x^2 + y^2 = r^2$ .
✓
Check point: substitute and verify.
✓
Tangent is perpendicular to the radius at the point of contact.

Tags:

#gcse#maths#algebra#circles#graphs

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Core Concepts

Standard Equation

Points on the Circle

Tangent to a Circle

Worked Example: Example 1

Worked Example: Example 2

Practice Problems

Key Takeaways

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