Knowledge organisers

Coordinate geometry: A Level Maths knowledge organiser

Everything to know about coordinate geometry on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Gradient: Change in y divided by change in x between two points on a line.
Perpendicular lines: Lines at right angles; their gradients multiply to −1.
Equation of a circle: (x − a)² + (y − b)² = r² has centre (a, b) and radius r.
Parametric equations: x and y are both given in terms of a third variable, the parameter t.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Line; gradient	\(y-y_1=m(x-x_1),\) \(m=\frac{y_2-y_1}{x_2-x_1}\)
Perpendicular gradients	\(m_1m_2=-1\)
Midpoint; distance	\(\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2\right),\) \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Circle, centre \((a,b)\), radius \(r\)	\((x-a)^2+(y-b)^2=r^2\)
Circles: tangent ⟂ radius; angle in a semicircle is 90°; perpendicular from centre bisects a chord
Parametric → Cartesian: eliminate \(t\), e.g. with \(\sin^2t+\cos^2t=1\)
Sphere; cone (curved surface, slant \(l\))In the formula booklet: Edexcel	\(A=4\pi r^2;\) \(A=\pi rl\)

Worked example

Find the centre and radius of the circle x² + y² − 4x + 6y − 12 = 0.

(x − 2)² − 4 + (y + 3)² − 9 − 12 = 0
(x − 2)² + (y + 3)² = 25

Answer: Centre (2, −3), radius 5

Common mistakes

F(x + a) moved in the +x direction
F(2x) stretched by 2 instead of ½
Asymptotes not transformed
Perpendicular gradient taken as −m

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Find the gradient and y-intercept of straight lines written in different forms, and rearrange equations such as 3x + 2y = 7 into the form y = mx + c.
Calculate the distance between two points and use straight-line equations and intersections to find lengths and areas of triangles and other shapes on coordinate axes.
Find the midpoint of a line segment and the equation of its perpendicular bisector, and use perpendicular bisectors of chords to locate the centre of a circle.
Use the facts that a tangent is perpendicular to the radius and that the perpendicular from the centre bisects a chord to find tangent equations and lengths.
Use parametric equations x = f(t) and y = g(t) to find points on a curve, and convert to a Cartesian equation by eliminating the parameter.
Find where parametric curves meet the axes or a line by solving for the parameter, and model paths such as projectiles with parametric equations.

The printable sheet

Coordinate geometry knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Coordinate geometry knowledge organiser (A Level Maths), A4. Download the PDF.

Revise it next

Other A Level Maths topics: Proof · Algebra and functions · Sequences and series · Trigonometry · Exponentials and logarithms · Differentiation · Integration · Numerical methods · Vectors · Sampling and data · Probability · Binomial and normal distributions · Hypothesis testing · Kinematics · Forces and Newton's laws · Moments · All A Level Maths organisers