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
(a) Write down the equation of the circle with centre $O$ and radius $9$ .

(1 mark)
(b) A different circle has equation $x^2 + y^2 = 0.25$

Write down the radius of this circle.

(1 mark)
Level 2 · 2 marksTry it
2
A circle has centre $(0, 0)$

The point $(-2, 6)$ lies on the circle.

Find an equation of the circle.

(2 marks)
Level 2 · 2 marksTry it
3
A circle has equation $x^2 + y^2 = 42$

Is the point $(5, -4)$ inside the circle, on the circle or outside the circle?

Tick one box.

☐ Inside the circle

☐ On the circle

☐ Outside the circle

Show working to support your answer.

(2 marks)
Level 2 · 2 marksTry it
4
The point $(k, 3)$ lies on the circle $x^2 + y^2 = 34$

$k$ is negative.

Find the value of $k$ .

(2 marks)
Level 2 · 2 marksTry it
5
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

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