Pair of Straight Lines
Easy Overview
You already know about a single straight line. Now imagine two lines passing through the same point — and we combine their equations into one. That's what this chapter is about: how to represent a pair of lines with a single equation, find the angle between them, and split them apart again.
Combined Equation of Two Lines
If you have two lines L₁ = 0 and L₂ = 0, their combined equation is L₁ × L₂ = 0. Multiply them together! For example, lines x − 2y = 0 and x + 3y = 0 combine into (x − 2y)(x + 3y) = 0, which gives x² + xy − 6y² = 0. The degree becomes 2 — that's why we call them 'pair of lines'.
Homogeneous Equations of Degree 2
An equation like ax² + 2hxy + by² = 0 is the general form of a pair of straight lines passing through the origin. Each term has degree 2 (x², xy, y²) — that's why it's 'homogeneous'. Think of it like both lines sharing the same parking spot (the origin).
Angle Between Lines
Given ax² + 2hxy + by² = 0, the angle θ between the two lines is: tan θ = |2√(h² − ab)| / |a + b|. If h² = ab, the lines are parallel (θ = 0). If a + b = 0, they're perpendicular (θ = 90°). One formula, two special cases. Easy.
Individual Lines from Combined Equation
To split the combined equation back into separate lines, just factor it. Treat ax² + 2hxy + by² = 0 like a quadratic in x (or y). Solve for x in terms of y, and you'll get two equations. Each one is a separate line. It's reverse-engineering the multiplication.
Key Points
- •Combined equation of lines L₁ = 0 and L₂ = 0 is L₁L₂ = 0
- •General pair through origin: ax² + 2hxy + by² = 0
- •Angle between lines: tan θ = |2√(h² − ab)| / |a + b|
- •If h² = ab → lines are parallel
- •If a + b = 0 → lines are perpendicular
- •Split the equation by factoring the quadratic in x or y
Practice Questions
- Find the combined equation of lines x + 2y = 0 and 2x − y = 0.
- Show that the equation x² − 5xy + 6y² = 0 represents two distinct lines and find the angle between them.
- If the lines represented by ax² + 2hxy + by² = 0 are perpendicular, prove that a + b = 0.
- Find the separate equations of the lines represented by x² + 4xy + 3y² = 0.