Learning Objectives
- Understand slope of a line and different forms of equations
- Learn about distance formulas and angle between lines
- Study the general equation of a line and family of lines
Key Concepts
Slope of a Line
Slope (m): m = tan θ (θ is angle with positive x-axis). m = (y₂ - y₁)/(x₂ - x₁) for two points. Horizontal line: m = 0. Vertical line: m = undefined. Parallel lines: m₁ = m₂. Perpendicular lines: m₁ × m₂ = -1.
Forms of Line Equation
Slope-intercept form: y = mx + c (c is y-intercept). Point-slope form: y - y₁ = m(x - x₁). Two-point form: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁). Intercept form: x/a + y/b = 1 (a = x-intercept, b = y-intercept). Normal form: x cos α + y sin α = p (p = perpendicular distance from origin, α = angle of normal). General form: Ax + By + C = 0 (slope = -A/B).
Distance Formulas
Distance between two points: d = √[(x₂-x₁)2 + (y₂-y₁)2].
Distance of point (x₁, y₁) from line Ax + By + C = 0: d = |Ax₁ + By₁ + C| / √(A2 + B2).
Distance between parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0: d = |C₁ - C₂| / √(A2 + B2).
Angle Between Two Lines
tan θ = |m₁ - m₂| / (1 + m₁m₂) where m₁, m₂ are slopes. If 1 + m₁m₂ = 0, lines are perpendicular. If m₁ = m₂, lines are parallel.
Family of Lines
Any line through the intersection of L₁ = 0 and L₂ = 0 is: L₁ + λL₂ = 0, where λ is a parameter. Useful for finding specific lines through intersection without explicitly finding the intersection point.
Important Results
Section formula: Point dividing line joining (x₁,y₁) and (x₂,y₂) in ratio m:n internally: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)). Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Area of triangle: (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|. Collinear points: Area of triangle = 0.
Summary
Straight lines are described by various equation forms. The slope determines the line's inclination. Key formulas include distance from a point to a line, angle between lines, and section formula. The family of lines concept simplifies problems involving intersection.
Important Terms
- Slope: Measure of steepness; m = tanθ
- Intercept form: x/a + y/b = 1
- Normal form: x cosα + y sinα = p
- Section formula: Divides a line segment in given ratio
- Collinear: Points lying on the same straight line
Quick Revision
- Parallel: m₁ = m₂; Perpendicular: m₁m₂ = -1
- Distance from point to line: |Ax₁+By₁+C|/√(A2+B2)
- Angle between lines: tanθ = |(m₁-m₂)/(1+m₁m₂)|
- Area of triangle = (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
- Intercept form: x/a + y/b = 1
- Family of lines: L₁ + λL₂ = 0
- Slope of Ax + By + C = 0 is -A/B