Working with Straight Lines – Coordinate Geometry (9709 Pure Mathematics 1) — Mathematics

Introduction

Straight-line geometry underpins almost every area of Pure Mathematics 1 and regularly appears in 9709 exam questions — both as a topic in its own right and as a tool inside curve-sketching, vectors, and calculus problems. The syllabus objective for this subtopic requires you to be fluent in three equivalent line forms, to compute distances, gradients, midpoints, and intersections, and to exploit the gradient conditions for parallel and perpendicular lines. Mastery here is non-negotiable for a high grade.

Core Concept

The Three Line Forms

Every straight line can be written in any of the following forms; you must recognise, convert between, and work with all three.

Form	Equation	Best used when…
Slope–intercept	$y = mx + c$	Reading off gradient $m$ and $y$ -intercept $c$ immediately
Point–slope	$y - y_1 = m(x - x_1)$	You know a point $(x_1, y_1)$ and gradient $m$
General (implicit)	$ax + by + c = 0$	Exam answers often require this; easy to check integer coefficients

Gradient

The gradient of a line through $(x_1, y_1)$ and $(x_2, y_2)$ measures steepness and direction:

m = \frac{y_2 - y_1}{x_2 - x_1}

Midpoint

The midpoint $M$ of the segment joining $(x_1, y_1)$ to $(x_2, y_2)$ is:

M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)

Distance

The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Parallel and Perpendicular Lines

Two lines are parallel if and only if their gradients are equal:

m_1 = m_2

Two lines are perpendicular if and only if their gradients satisfy:

m_1 \times m_2 = -1 \quad \Longleftrightarrow \quad m_2 = -\frac{1}{m_1}

This is one of the most frequently tested relationships on the 9709 paper.

Points of Intersection

The intersection of two lines is found by solving their equations simultaneously — either by substitution or elimination.

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