A-Level Maths: Coordinate Geometry

Coordinate geometry is where A-Level algebra meets shape, and its exam questions are built from a small set of reliable moves: gradients, perpendicularity, completing the square, and the circle's geometric properties translated into equations. This quiz drills each of them with concrete coordinates throughout.

The straight-line questions cover the core workflow: finding the equation of a line through two points (gradient first, then the intercept by substitution), identifying which lines are parallel to 3x + 2y = 7 — a question that forces rearrangement before gradients can be compared, including the disguised multiple 6x + 4y = 1 — and constructing the perpendicular through a given point using the negative reciprocal rule on a line given in implicit form.

The circle questions carry the topic's A-Level weight: completing the square on x² + y² − 6x + 8y = 0 to extract centre (3, −4) and radius 5, verifying which points lie on a given circle by substitution, and the tangent question that uses the defining property — tangent perpendicular to radius at the point of contact — to derive 3x + 4y = 25 at the point (3, 4). A statement-checking question on (x − 2)² + (y − 3)² = 16 tests reading the equation precisely: the centre by inspection, the radius as the square root of 16, and whether the circle reaches the x-axis (it does, since the centre sits 3 units up with radius 4).

Parametric equations close the set at introductory level: verifying points against x = 2t, y = t², and eliminating the parameter from x = 3cos t, y = 3sin t via the Pythagorean identity to recover x² + y² = 9.

Every key has been checked by substitution.

• Find equations of lines through given points and identify parallel lines in any form
• Construct perpendicular lines using the negative reciprocal rule
• Complete the square to find a circle's centre and radius
• Use the tangent-radius property to find tangent equations
• Convert between parametric and Cartesian forms in simple cases

Topic scope follows section C (Coordinate geometry in the (x,y) plane) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: straight lines, circle equations and properties, and parametric equations.