GCSE · Circles, arcs & sectors

Equation of a circle (x² + y² = r²)Exam-style questions

Why every point of a circle centred at the origin obeys x² + y² = r² (it is Pythagoras), how to read the radius off an equation and describe the graph fully, how to test whether a point is inside, on or outside, how to find the tangent at a point and where it meets an axis, where a line crosses the circle, and how to sketch the circle after a translation.

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19 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The diagram shows the plan of a garden on a coordinate grid. One unit on each axis represents $1$ metre.

A sprinkler at the origin $O$ waters every point that is up to $4.5$ m from $O$ .

The circle shows the edge of the watered region.(a) Write down the equation of the circle.

(2 marks)
(b) A rose bush is planted at the point $R(2.8, 3.6)$

Does the sprinkler water the rose bush?

Show how you decide.

(3 marks)
Level 3 · 5 marksTry it
2
(a) Describe fully the graph of $x^2 + y^2 = 72$

(3 marks)
(b) Write down the coordinates of the two points where the graph crosses the $x$ -axis.

Give your answers in their simplest form.

(1 mark)
Level 3 · 4 marksTry it
3
Here is a question and an incorrect solution.

Question: The point $P(2, -5)$ lies on the circle $x^2 + y^2 = 29$ . Work out the gradient of the tangent to the circle at $P$ .

Solution: The gradient of $OP$ is $\dfrac{-5}{2}$ , so the gradient of the tangent at $P$ is $-\dfrac{5}{2}$ .
(a) Describe the error made.

(1 mark)
(b) Work out the correct gradient of the tangent at $P$ .

(2 marks)
Level 3 · 3 marksTry it
4
$A$ and $B$ are two points on the circle $x^2 + y^2 = 89$

$A$ and $B$ both have $x$ coordinate $5$

Work out the length of $AB$ .

(3 marks)
Level 3 · 3 marksTry it
5
The point $P(2, 3)$ lies on the circle $x^2 + y^2 = 13$
(a) Work out the gradient of the radius $OP$ .

(1 mark)
(b) Write down the gradient of the tangent to the circle at $P$ .

(1 mark)
(c) Find the equation of the tangent to the circle at $P$ .

Give your answer in the form $ax + by = c$ , where $a$ , $b$ and $c$ are integers.

(2 marks)
Level 3 · 4 marksTry it