Coordinate geometry

Edexcel Pure 3

A-level Maths (9MA0) · exam-style practice, examiner-report intelligence and the tools that drill it.

Board and spec code confirmed against Edexcel 9MA0 · registry checked 2026-07-11How this checking works

Edexcel 9MA0 revision notes Edexcel 9MA0 past papers

The topic on one screen

Circle equation: $(x - a)^2 + (y - b)^2 = r^2$ . Given the expanded form, complete the square in $x$ and $y$ to extract centre and radius.
Perpendicular gradients multiply to $-1$ ; parallel lines share a gradient. Most line questions are one of these two facts.
A tangent to a circle is perpendicular to the radius at the point of contact — that's the whole method.
Line meets circle: substitute and read the discriminant. Two roots = crosses, repeated root = tangent, no roots = misses.
For centres/radii in terms of a constant $k$ , the radius must be real and positive: $r^2 > 0$ gives the condition on $k$ .
Distance between points: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ — leave surds exact unless told otherwise.

Where students actually lose marks

In the circle-with-k question the centre and $r^2$ must be EXTRACTED from the completed-square form — quoting $r^2$ still buried inside the equation doesn't score, and $\pm\sqrt{\ldots}$ for a radius is not allowed.

June 2023 Paper 1 mark scheme (Q10)

The scheme condones arithmetic slips inside the rearrangement but requires an attempted $b^2 - 4ac$ with at least one critical value for the intersection condition — a bare inequality with no discriminant work scores nothing.

June 2023 Paper 1 mark scheme (Q10)

Radius answers that then get 'simplified' further (e.g. divided by 2 after the square root) lose the mark — schemes do not apply isw when the extra step changes the value.

June 2023 Paper 1 mark scheme (Q10)

Try it — exam-style

Hard

ORIGINAL · after June 2023 Paper 1 Q10

A circle has equation

x^2 + y^2 - 4kx + 6ky + 9 = 0

, where

k

is a constant. Find, in terms of

k

, (i) the centre and (ii) the radius, and (iii) state the condition on

k

for the circle to exist.

(5)

(Total for Question 1 is 5 marks)

Medium

ORIGINAL

The circle

(x - 3)^2 + (y + 2)^2 = 25

passes through the point

P(6, 2)

. Find the equation of the tangent to the circle at

P

(4)

(Total for Question 2 is 4 marks)

Easy

ORIGINAL

Line

L_1

has equation

y = 3x - 7

. Line

L_2

passes through

(6, 1)

and is perpendicular to

L_1

. Find the equation of

L_2

(3)

(Total for Question 3 is 3 marks)

Medium

ORIGINAL

A(1, 5)

and

B(7, -3)

are the ends of a diameter of a circle. Find the equation of the circle.

(3)

(Total for Question 4 is 3 marks)

Questions are written in the style of past Edexcel papers (source shown on each) — never copied from them.

Get the printable question packbrowser print view

Drill it properly

Circle Theorems

The GCSE circle facts A-level questions quietly assume.

Algebra & functions

The completing-the-square skill this topic runs on.

Maths flashcards

Circle forms and gradient rules, on a review schedule.

Stuck on coordinate geometry?

Circle questions are three ideas wearing different costumes — I show you the costumes. Free intro call, then a free first lesson.

Book a free intro call