Various Forms of the Equation of a Line- Straight Lines

Straight Lines

9.1 Introduction

A straight line is the shortest distance between two points and is fundamental in coordinate geometry. The study of straight lines involves understanding their slope, various forms of equations, and the calculation of distances related to lines. These concepts are essential for solving problems in geometry, calculus, and physics.

9.2 Slope of a Line

The slope (m) of a line is a measure of its steepness and is defined as the ratio of the change in y-coordinate to the change in x-coordinate between any two points on the line.

• Formula: If the line passes through points (x₁, y₁) and (x₂, y₂):
  (y₂ − y₁) (x₂ − x₁) , or written as:
  m = y₂ − y₁x₂ − x₁
• If the line makes an angle θ with the positive direction of the x-axis, then
  m = tan(θ)
• Special cases:
  • Horizontal line: m = 0
  • Vertical line: m is undefined

9.3 Various Forms of the Equation of a Line

There are several standard forms to represent the equation of a straight line in the coordinate plane:

• 1. Slope-Point Form:
  Equation of a line with slope m passing through point (x₁, y₁):
  y − y₁ = m(x − x₁)
• 2. Two-Point Form:
  Equation of a line passing through points (x₁, y₁) and (x₂, y₂):
  y − y₁ = y₂ − y₁ x₂ − x₁ (x − x₁)
• 3. Slope-Intercept Form:
  Equation of a line with slope m and y-intercept c:
  y = mx + c
• 4. Intercept Form:
  If a line cuts the x-axis at a and the y-axis at b:
  x a + y b = 1
• 5. General Form:
  ax + by + c = 0
• 6. Normal Form:
  Equation of a line at a perpendicular distance l from the origin and making an angle β with the positive x-axis:
  x cos(β) + y sin(β) = l
• 7. Equations of Horizontal and Vertical Lines:
  • Horizontal (parallel to x-axis): y = a
  • Vertical (parallel to y-axis): x = a

Example: Equation through Two Points

Find the equation of the line passing through (-2, 4) and (1, 2):
Slope, 2 − 4 1 − (−2) = −2 3
Using point-slope form: y − 4 = −2 3 (x + 2)
Rearranged: 2x + 3y − 8 = 0

9.4 Distance of a Point From a Line

The perpendicular distance (d) from a point (x₁, y₁) to the line ax + by + c = 0 is given by:

d = |a x₁ + b y₁ + c| √(a² + b²)

Example: Find the distance from the point (3, −2) to the line 2x − 3y + 5 = 0:
d = |2×3 − 3×(−2) + 5| √(2² + (−3)²) = 17 √13

Summary Table: Forms of the Equation of a Line

Form	Equation	Description
Slope-Point	y − y₁ = m(x − x₁)	Line with slope m through (x₁, y₁)
Two-Point	y − y₁ = y₂ − y₁ x₂ − x₁ (x − x₁)	Line through (x₁, y₁) and (x₂, y₂)
Slope-Intercept	y = mx + c	Slope m, y-intercept c
Intercept Form	x a + y b = 1	x-intercept a, y-intercept b
General Form	ax + by + c = 0	Standard linear equation
Normal Form	x cos(β) + y sin(β) = l	Perpendicular distance l from origin, angle β
Horizontal	y = a	Parallel to x-axis
Vertical	x = a	Parallel to y-axis