1. Slope of a Line
The tangent of the angle θ made by a line with the positive direction of the x-axis is called its slope (m = tan θ).
Given two points (x₁, y₁) and (x₂, y₂), Slope m = (y₂ - y₁) / (x₂ - x₁).
Parallel and Perpendicular Lines
Parallel lines: m₁ = m₂ (Slopes are equal)
Perpendicular lines: m₁ × m₂ = -1
2. Various Forms of the Equation of a Line
- Point-Slope Form: y - y₁ = m(x - x₁)
- Two-Point Form: y - y₁ = [(y₂ - y₁)/(x₂ - x₁)] (x - x₁)
- Slope-Intercept Form: y = mx + c (where c is y-intercept)
- Intercept Form: x/a + y/b = 1 (a and b are x and y intercepts)
- Normal Form: x cos ω + y sin ω = p (p is perpendicular distance from origin, ω is angle of perpendicular)
3. Distance of a Point from a Line
d = |Ax₁ + By₁ + C| / √(A² + B²)
The perpendicular distance from point (x₁, y₁) to the line Ax + By + C = 0.
Distance between two parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 is d = |C₁ - C₂| / √(A² + B²).