IAL P2 Circles: Exam Questions & Answers

P2 Circles notes

The equation of a circle

$(x - a)^{2} + (y - b)^{2} = r^{2}$ . From an expanded form, complete the square in $x$ and in $y$ : $x^{2} + y^{2} + 8 x - 2 y - 8 = 0$ becomes $(x + 4)^{2} + (y - 1)^{2} = 25$ , so the centre is $(- 4, 1)$ and the radius is $5$ .

Tangents

A tangent is perpendicular to the radius at the point of contact. For the circle with centre $(1, 2)$ through $(4, 6)$ , the radius has gradient $\frac{4}{3}$ , so the tangent at $(4, 6)$ has gradient $- \frac{3}{4}$ : $y - 6 = - \frac{3}{4} (x - 4)$ .

Chords and circle geometry

The perpendicular bisector of any chord passes through the centre, so two chords' bisectors meet at the centre. The angle in a semicircle is a right angle: if $A B$ is a diameter and $C$ is on the circle, $∠ A C B = 90^{\circ}$ , so the gradients of $A C$ and $B C$ multiply to $- 1$ .

Where a line meets a circle

Substitute the line into the circle to get a quadratic. Two roots: the line cuts the circle; one repeated root: it is a tangent; no real roots: it misses. For $x^{2} + y^{2} = 10$ and $y = x + 2$ : $2 x^{2} + 4 x - 6 = 0$ , so $x = 1$ or $x = - 3$ , and the line meets the circle at $(1, 3)$ and $(- 3, - 1)$ .

Key formulas

Circle Learn it: $(x - a)^{2} + (y - b)^{2} = r^{2}; tangent ⊥ radius$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P2 formulas on one page (free PDF) →

P2 Circles worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 7 marks · hard

The points $A (1, 2)$ and $B (7, 10)$ lie on a circle $C$ with $A B$ as a diameter.
(a) Find the equation of $C$ . (4)
(b) The tangent to $C$ at $A$ meets the $y$ -axis at $T$ . Find the coordinates of $T$ . (3)

Mark scheme

(a) Centre $=$ midpoint $= (4, 6)$ B1. Radius $= \frac{1}{2} | A B | = \frac{1}{2} \sqrt{36 + 64} = 5$ M1 A1. Equation: $(x - 4)^{2} + (y - 6)^{2} = 25$ A1.
(b) Radius from centre to $A$ has gradient $\frac{2 - 6}{1 - 4} = \frac{4}{3}$ B1. Tangent gradient $= - \frac{3}{4}$ ; equation: $y - 2 = - \frac{3}{4} (x - 1)$ . Set $x = 0$ : $y = 2 + \frac{3}{4} = \frac{11}{4}$ M1 A1.

Worked example 2 · 6 marks · hard

The circle $C$ has centre $(a, b)$ and passes through the points $P (0, 0)$ , $Q (6, 0)$ , and $R (0, 8)$ . Find the values of $a$ , $b$ , and the radius of $C$ .

Mark scheme

$P$ and $Q$ both on circle ⇒ centre lies on the perpendicular bisector of $P Q$ , so $a = 3$ M1 A1. Similarly $P$ and $R$ give $b = 4$ M1 A1. Radius $= | C P | = \sqrt{9 + 16} = 5$ M1 A1.

Worked example 3 · 5 marks · medium

The tangent to the circle $x^{2} + y^{2} = 25$ at the point $P (3, 4)$ meets the $x$ -axis at $Q$ . Find the coordinates of $Q$ .

Mark scheme

Radius $O P$ has gradient $\frac{4}{3}$ B1. Tangent gradient = $- \frac{3}{4}$ (perpendicular to radius) M1. Equation: $y - 4 = - \frac{3}{4} (x - 3)$ A1. Set $y = 0$ : $- 4 = - \frac{3}{4} (x - 3) \Rightarrow x - 3 = \frac{16}{3}$ M1. So $x = \frac{25}{3}$ A1.

Common P2 circles mistakes

Sign of the centre. $(x + 4)^{2}$ means the centre's $x$ -coordinate is $- 4$ , not $4$ .
Radius squared. In $(x - a)^{2} + (y - b)^{2} = 25$ the radius is $5$ , not $25$ .
Tangent gradient. Use the negative reciprocal of the radius's gradient, not the radius's gradient itself.
Completing the square once. Complete the square in x and in y, and move both constants to the right-hand side.

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P2 Circles FAQ

How do I find the centre and radius of a circle?

Complete the square in x and in y to get (x − a)² + (y − b)² = r². The centre is (a, b) and the radius is the square root of the number on the right.

How do I show a line is a tangent to a circle?

Substitute the line into the circle's equation and show the quadratic has a repeated root (discriminant zero), or show that the distance from the centre to the line equals the radius.

Is a calculator allowed in IAL P2?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P2 included. Method marks are for the working you write down, so show it.

How long is the IAL P2 exam?

Each IAL maths unit, P2 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does circles come up in P2?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, circles was tested in 4 of them (4 questions in all). Any topic can be examined in any series.

Next steps

More P2: Sequences and Series · Logarithms and Exponentials · Further Trigonometry · Algebraic Methods · Proof · The Binomial Expansion · Differentiation (stationary points) · Integration (definite integrals and area)

Edexcel IAL P2 Circles: exam questions and answers, notes and worked examples

Try it first

P2 Circles notes

The equation of a circle

Tangents

Chords and circle geometry

Where a line meets a circle

Key formulas

P2 Circles worked examples

Worked example 1 · 7 marks · hard

Mark scheme

Worked example 2 · 6 marks · hard

Mark scheme

Worked example 3 · 5 marks · medium

Mark scheme

Common P2 circles mistakes

P2 Circles FAQ

How do I find the centre and radius of a circle?

How do I show a line is a tangent to a circle?

Is a calculator allowed in IAL P2?

How long is the IAL P2 exam?

How often does circles come up in P2?

Next steps

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