Coordinate Geometry

This lesson covers Coordinate Geometry as required by the A-Level Mathematics Pure 1 specification. Coordinate geometry connects algebra with geometry by using equations to describe lines, curves, and circles on a Cartesian plane. This topic is fundamental and appears throughout the A-Level course.

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Equations of Straight Lines

Gradient (Slope)

The gradient of a line passing through two points (x₁, y₁) and (x₂, y₂) is:

m = (y₂ − y₁) / (x₂ − x₁)

Forms of a Straight Line Equation

Form	Equation	Use
Gradient-intercept	y = mx + c	When gradient and y-intercept are known
Point-gradient	y − y₁ = m(x − x₁)	When gradient and a point are known
General form	ax + by + c = 0	Standard form for answers

Example: Find the equation of the line through (2, 5) with gradient 3.

y − 5 = 3(x − 2)
y − 5 = 3x − 6
y = 3x − 1

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Parallel and Perpendicular Lines