Mathematics 9709 · syllabus 1.3
Cambridge A Level Mathematics 9709 — Coordinate geometry topical past papers
Straight lines, gradients, midpoints and the equation of a circle, and how the points where two graphs meet give the solutions of equations.
- Questions
- 110
- Years
- 2009–2026
- With mark scheme
- 98%
- Questions 2022–2026
- 46
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Coordinate geometry covers
Coordinate geometry joins algebra to pictures. For straight lines you find gradients, midpoints and distances, write equations in the forms y = mx + c, y − y₁ = m(x − x₁) or ax + by + c = 0, and use the rule that the gradients of perpendicular lines multiply to −1. Typical tasks are finding a perpendicular bisector, the foot of a perpendicular, or the point where two lines cross.
Circles are the second half. The equation (x − a)² + (y − b)² = r² describes the circle with centre (a, b) and radius r; given the expanded form x² + y² + 2gx + 2fy + c = 0, you complete the square twice to find the centre and radius. Problems use the facts that a tangent is perpendicular to the radius at the point of contact, that the angle in a semicircle is a right angle, and that the perpendicular bisector of a chord passes through the centre.
The topic is section 1.3 of Paper 1. Where a line meets a circle or another curve, substitution gives a quadratic, so the discriminant from quadratics decides whether they cross, touch or miss. Implicit differentiation is not expected here; tangents to circles come from the radius instead. Later, differentiation produces tangents and normals to other curves as straight-line equations, and in Paper 3 the circle and the perpendicular bisector return as loci of complex numbers on an Argand diagram.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 1.3 Coordinate geometry
What you should be able to do
- Find the equation of a line from two points, or from a point and a gradient
- Use gradients to decide whether lines are parallel or perpendicular
- Calculate distances, midpoints and points of intersection
- Find the centre and radius of a circle from its equation in either form
- Find the equation of a tangent to a circle at a given point
- Decide whether a line cuts, touches or misses a circle or curve
Key ideas
- Gradient m = (y₂ − y₁)/(x₂ − x₁); the line through (x₁, y₁) with gradient m is y − y₁ = m(x − x₁)
- Perpendicular lines have m₁m₂ = −1; parallel lines have equal gradients
- Distance = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
- (x − a)² + (y − b)² = r² has centre (a, b) and radius r; x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c)
- A tangent to a circle is perpendicular to the radius at the point of contact
How Coordinate geometry has been examined
From 2022 to 2026, Coordinate geometry was on Paper 1 · Pure Mathematics 1 in 11 of 11 exam series, worth about 15% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Pure Mathematics 1 | 46 | 15% | 11 of 11 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 8.
Filed in Momentum under: Coordinates Geometry.
A worked example
A circle has equation x² + y² − 6x + 4y − 12 = 0. Find its centre and radius, and the equation of the tangent to the circle at the point (6, 2).
- Complete the square in x and in y: (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0.
- So (x − 3)² + (y + 2)² = 25: the centre is (3, −2) and the radius is 5.
- Check that (6, 2) lies on the circle: (6 − 3)² + (2 + 2)² = 9 + 16 = 25.
- The radius to (6, 2) has gradient (2 − (−2))/(6 − 3) = 4/3, so the tangent, perpendicular to it, has gradient −3/4.
- y − 2 = −¾(x − 6), so 4y − 8 = −3x + 18, which rearranges to 3x + 4y − 26 = 0.
Answer: Centre (3, −2), radius 5; tangent 3x + 4y − 26 = 0
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Reading the centre of (x + 3)² + (y − 1)² = 20 as (3, −1): the signs flip, so the centre is (−3, 1).
- Giving the radius of that circle as 20: the right-hand side is r², so r = √20 = 2√5.
- Using the gradient of the radius as the gradient of the tangent: the tangent's gradient is the negative reciprocal.
- Drawing a perpendicular bisector through one end of a segment rather than its midpoint: a bisector must pass through the midpoint.
Exam technique
- Draw a rough sketch with the given points; it catches sign errors and shows which of two intersection points a question means.
- Leave lengths in surd form, such as 2√5, unless a decimal is asked for, and give a line's equation in the form requested, for example ax + by + c = 0 with integer coefficients.
- For a line meeting a circle, substitute the line into the circle's equation and use the discriminant of the resulting quadratic.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.