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
$P(4, -6)$ is a point on the circle $x^2 + y^2 = 52$

The tangent to the circle at $P$ meets the $x$ -axis at $A$ and the $y$ -axis at $B$ .

Kai says,

" $A$ is further from the origin than $B$ is."

Is Kai correct?

You must show how you get your answer.

(4 marks)
Level 5 · 4 marksTry it
2
The points $A(1, 7)$ and $B$ both lie on a circle with centre $O(0, 0)$ .

The $y$ -coordinate of $B$ is $-5$

The gradient of the line $AB$ is negative.

Work out the exact length of $AB$ .

Give your answer in its simplest form.

(5 marks)
Level 5 · 5 marksTry it
3
The diagram shows the circle $x^2 + y^2 = 80$ and the point $P(-8, 4)$ on the circle.

The tangent to the circle at $P$ meets the $y$ -axis at $S$ .Not to scale

Work out the exact length of $PS$ .

Give your answer in the form $a\sqrt{b}$ , where $a$ and $b$ are integers and $b$ is as small as possible.

You must show your working.

(5 marks)
Level 5 · 5 marksTry it