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
Here is a circle, centre $O$ .

The point $P(-6, 1)$ lies on the circle.

The line $T$ is the tangent to the circle at $P$ .Not drawn accurately

Work out the equation of $T$ .

Give your answer in the form $y = mx + c$

You must show your working.

(4 marks)
Level 4 · 4 marksTry it
2
The diagram shows the circle $x^2 + y^2 = 58$ and the point $P(3, 7)$ on the circle.

The tangent to the circle at $P$ crosses the $x$ -axis at the point $T$ .Not to scale
(a) Show that $T$ has coordinates $\left(\dfrac{58}{3}, 0\right)$ .

(4 marks)
(b) Work out the exact area of triangle $OPT$ .

(2 marks)
Level 4 · 6 marksTry it
3
The line $L$ has equation $2x + 5y = 58$

$L$ is a tangent to a circle whose centre is the origin.

$L$ touches the circle at the point $P$ .
(a) Work out the coordinates of $P$ .

(4 marks)
(b) Write down the equation of the circle.

(1 mark)
Level 4 · 5 marksTry it
4
The line $y = x + 3$ meets the circle $x^2 + y^2 = 65$ at the points $A$ and $B$ .
(a) Use an algebraic method to find the coordinates of $A$ and $B$ .

You must show your working.

(5 marks)
(b) Work out the exact length of $AB$ .

Give your answer in the form $k\sqrt{2}$ , where $k$ is an integer.

(2 marks)
Level 4 · 7 marksTry it
5
Circle $C_1$ has equation $x^2 + y^2 = 49$

A translation by the vector $\begin{pmatrix} 5 \\ 0 \end{pmatrix}$ maps $C_1$ onto a circle $C_2$ .
(a) Write down the coordinates of the centre of $C_2$ .

(1 mark)
(b) Sketch $C_2$ on a pair of axes.

Label the coordinates of each point where $C_2$ crosses the $x$ -axis.

(2 marks)
(c) $C_2$ crosses the $y$ -axis at two points.

Find the exact coordinates of these points.

Give your answers in their simplest form.

(3 marks)
Level 4 · 6 marksTry it

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