Distance Between Parallel Lines

Distance Between Parallel Lines

Introduction:

  • The distance between parallel lines refers to the shortest distance measured perpendicular from any point on one line to the other parallel line.

Formula for Distance between Parallel Lines:

  • Given two parallel lines:
    1. Line 1: Ax+By+C1=0
    2. Line 2: Ax+By+C2=0
  • The distance 'd' between these parallel lines is given by:
    Distance=C2C1A2+B2

Steps to Find Distance:

  1. Calculate Numerator:
    • Subtract the constants (intercepts) of the equations of the parallel lines: C2C1.
  2. Calculate Denominator:
    • Use the coefficients of x and y in the line equations to find A2+B2
    • .
  3. Divide Numerator by Denominator:
    • Divide C2C1 by A2+B2
    • to determine the distance.

Example:

Given two parallel lines:

  1. Line 1: 3x+4y5=0

  2. Line 2: 3x+4y+7=0

  3. Calculate Numerator:

    • C2C1=7(5)=12
  4. Calculate Denominator:

    • Use coefficients of x and y: 32+42=9+16=25=5
  1.  
  2. Find Distance:

    • Distance =125=125=2.4 units

Importance:

  • Determines the shortest distance between two parallel lines.
  • Crucial in geometry, architecture, and engineering for accurate measurements and spatial planning.

Summary:

  • The distance between parallel lines is computed using the difference in their constants divided by the square root of the sum of the squares of their coefficients.
  • This measurement provides critical information for spatial analysis, optimization, and design in various fields.