Straight Lines

Concept overview

A straight line is fully determined by any two points on it, or by one point and a slope. Slope m = (y₂ − y₁)/(x₂ − x₁). Several equivalent equation forms exist: point-slope y − y₁ = m(x − x₁); slope-intercept y = mx + c; two-point (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁); intercept x/a + y/b = 1; normal x cos α + y sin α = p (α is angle of normal from x-axis, p is distance from origin).

Angle between two lines with slopes m₁, m₂ is tan θ = |(m₁ − m₂)/(1 + m₁ m₂)|. Lines are parallel iff m₁ = m₂; perpendicular iff m₁ m₂ = −1.

Distance from point (x₀, y₀) to line ax + by + c = 0 is |ax₀ + by₀ + c|/√(a² + b²). Distance between two parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0 is |c₁ − c₂|/√(a² + b²).

Section formula for point dividing (x₁, y₁) and (x₂, y₂) in ratio m:n is ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)). Centroid, orthocentre, incentre and circumcentre of a triangle are the classic derived points.

Key formulas & concepts

• Slope: m = (y₂ − y₁)/(x₂ − x₁)
• Point-slope: y − y₁ = m(x − x₁)
• General form: ax + by + c = 0
• Distance from point to line: |ax₀ + by₀ + c|/√(a² + b²)
• Angle between lines: tan θ = |(m₁ − m₂)/(1 + m₁m₂)|
• Section formula: ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))
• Perpendicular condition: m₁ × m₂ = −1

Common mistakes to avoid

• Using slope formula for vertical lines — slope is undefined for x₁ = x₂.
• Ignoring the ± when solving angle-between-lines equations.
• Confusing internal and external division in the section formula.
• Forgetting the absolute value in distance formulas.

JEE Main relevance

JEE Main asks 2 straight-line problems every paper — usually one distance/angle calculation and one triangle-related coordinate problem.

JEE Advanced relevance

Advanced problems combine straight lines with locus problems or use trigonometric identities for angle bisectors and family-of-lines problems.

Previous-year questions (PYQ)

Angle between lines and distance from point to line dominate the PYQ pool.