Straight Lines

Equation of a Straight Line

• General form: ax + by + c = 0
• Equation of x-axis: y = 0
• Equation of y-axis: x = 0
• Line parallel to x-axis at distance a: y = a
• Line parallel to y-axis at distance a: x = a

Slope of a Line

• Slope (m) = tanθ, where θ is the angle with positive x-axis
• Slope of line joining (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁)/(x₂ − x₁)
• Slope of ax + by + c = 0: m = −a/b
• Slope of x-axis = 0
• Slope of y-axis = ∞

Standard Forms of Straight Line

Slope–Intercept Form

• y = mx + c
• m = slope, c = y-intercept

Point–Slope Form

• y − y₁ = m(x − x₁)

Two-Point Form

• y − y₁ = [(y₂ − y₁)/(x₂ − x₁)](x − x₁)

Intercept Form

• x/a + y/b = 1
• a, b are x- and y-intercepts respectively

Normal (Perpendicular) Form

• x cosα + y sinα = p
• p = perpendicular distance from origin

Parametric Form

• Line through (x₁, y₁) making angle θ with x-axis:
• x = x₁ + r cosθ
• y = y₁ + r sinθ

Reduction of General Form

• Slope–intercept form: y = −(a/b)x − (c/b)
• Intercepts: x-intercept = −c/a, y-intercept = −c/b
• Normal form: ax/√(a²+b²) + by/√(a²+b²) = −c/√(a²+b²)

Position of a Point Relative to a Line

• Point (x₁, y₁) lies on line if ax₁ + by₁ + c = 0
• Two points lie on same side if expressions have same sign
• They lie on opposite sides if expressions have opposite signs

Angle Between Two Lines

• If slopes are m₁, m₂:
• tanθ = |(m₁ − m₂)/(1 + m₁m₂)|
• Parallel lines: m₁ = m₂
• Perpendicular lines: m₁m₂ = −1

Parallel & Perpendicular Lines

• Line parallel to ax + by + c = 0: ax + by + k = 0
• Line perpendicular to it: bx − ay + k = 0

Distance Between Two Parallel Lines

• Between ax + by + c₁ = 0 and ax + by + c₂ = 0:
• d = |c₁ − c₂| / √(a² + b²)

Angle Bisectors of Two Lines

• For lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
• (a₁x + b₁y + c₁)/√(a₁² + b₁²) = ± (a₂x + b₂y + c₂)/√(a₂² + b₂²)

Lines Through Intersection of Two Lines

• If lines are L₁ = 0 and L₂ = 0:
• Family of lines: L₁ + λL₂ = 0

Pair of Straight Lines (Homogeneous Equation)

• ax² + 2hxy + by² = 0 represents a pair of lines through origin if h² ≥ ab
• Slopes m₁, m₂:
  • m₁ + m₂ = −2h/b
  • m₁m₂ = a/b
• Perpendicular if a + b = 0
• Identical if h² = ab

General Second Degree Equation

• ax² + 2hxy + by² + 2gx + 2fy + c = 0
• Represents a pair of straight lines if:
• abc + 2fgh − af² − bg² − ch² = 0

Foot of Perpendicular & Reflection

• Foot of perpendicular from (x₁, y₁) to ax + by + c = 0:
• (x₁ − a(ax₁ + by₁ + c)/(a²+b²), y₁ − b(ax₁ + by₁ + c)/(a²+b²))
• Reflection point obtained by doubling the perpendicular distance