A-Level Maths: Coordinate Geometry Revision

Coordinate Geometry is a core part of A-Level Maths. Revise the key concepts and common mistakes below, then lock them in with the free games.

Key concepts

Gradientm = (y₂ − y₁)/(x₂ − x₁): the change in y divided by the change in x between two points on a line.

Equation of a straight lineUse y − y₁ = m(x − x₁) for a line through (x₁, y₁) with gradient m, or y = mx + c with y-intercept c.

Parallel and perpendicular linesParallel lines have equal gradients; perpendicular gradients multiply to −1, so m₂ = −1/m₁.

Midpoint and distanceThe midpoint of two points is ((x₁ + x₂)/2, (y₁ + y₂)/2); the distance between them is √((x₂ − x₁)² + (y₂ − y₁)²).

Equation of a circle(x − a)² + (y − b)² = r² is the circle with centre (a, b) and radius r.

Completing the square for circlesRewrite x² + y² + 2gx + 2fy + c = 0 by completing the square in x and y: centre (−g, −f), radius √(g² + f² − c).

Circle propertiesThe perpendicular from the centre to a chord bisects it, a tangent is perpendicular to the radius, and the angle in a semicircle is 90°.

Line and circle intersectionsSubstitute the line into the circle; the discriminant of the quadratic shows two intersections (> 0), a tangent (= 0) or none (< 0).

Parametric equationsx and y are each given in terms of a parameter, such as x = f(t), y = g(t), tracing out a curve as t varies.

Parametric to CartesianEliminate the parameter, by substitution or an identity such as sin²t + cos²t = 1, to get a single equation linking x and y.