Distance between two parallel lines


Given are two parallel straight lines with slope m, and different y-intercepts b1 & b2.The task is to find the distance between these two parallel lines.

Examples:

Input: m = 2, b1 = 4, b2 = 3
Output: 0.333333

Input: m = -4, b1 = 11, b2 = 23
Output: 0.8


Approach:



  1. Let PQ and RS be the parallel lines, with equations
    y = mx + b1
    y = mx + b2

  2. The distance between these two lines is the distance between the two intersection points of these lines with the perpendicular line.Let that distance be d.
  3. So, equation of the line perpendicular to PQ and RS can be
    y = -x/m
  4. Now, solving the perpendicular line with PQ and RS separately to get the intersecting points (x1, y1) & (x2, y2), we get,
  5. From PQ,
    y = mx + b1
    y = -x/m
    (x1, y1) = ( -b1*m/(m^2 + 1), b1/(m^2 + 1))
  6. From RS,
    y = mx + b2
    y = -x/m
    (x2, y2) = ( -b2*m/(m^2 + 1), b2/(m^2 + 1))
  7. So, d = distance between (x1, y1) and (x2, y2)
     d = \frac{\left | b2 - b1 \right |}{ sqrt( m^{2} + 1 ) }

Below is the implementation of the above approach:

C++

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// C++ program find the distance
// between two parallel lines
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the distance 
// between parallel lines 
double dist(double m, double b1, double b2)
{
    double d = fabs(b2 - b1) / ((m * m) - 1);
    return d;
}
  
// Driver Code
int main()
{
    double m = 2, b1 = 4, b2 = 3;
    cout << dist(m, b1, b2);
    return 0;
}

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Java

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// Java program find the distance
// between two parallel lines
class GFG
{
      
// Function to find the distance 
// between parallel lines 
static double dist(double m, 
                double b1, double b2)
{
    double d = Math.abs(b2 - b1) / 
                    ((m * m) - 1);
    return d;
}
  
// Driver Code
public static void main(String[] args)
{
    double m = 2, b1 = 4, b2 = 3;
     System.out.println(dist(m, b1, b2));
}
}
  
// This code is contributed by Code_Mech.

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Python3

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# Python3 program find the distance
# between two parallel lines
  
# Function to find the distance 
# between parallel lines 
def dist(m, b1, b2):
    d = abs(b2 - b1) / ((m * m) - 1);
    return d;
  
# Driver Code
def main():
    m, b1, b2 =2,4, 3;
    print(dist(m, b1, b2));
if __name__ == '__main__':
    main()
  
# This code contributed by PrinciRaj1992

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C#

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// C# program find the distance
// between two parallel lines
using System;
  
class GFG
{
      
// Function to find the distance 
// between parallel lines 
static double dist(double m, 
                   double b1, double b2)
{
    double d = Math.Abs(b2 - b1) / 
                      ((m * m) - 1);
    return d;
}
  
// Driver Code
public static void Main()
{
    double m = 2, b1 = 4, b2 = 3;
    Console.Write(dist(m, b1, b2));
}
}
  
// This code is contributed by Akanksha Rai

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PHP

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<?php
// PHP program find the distance 
// between two parallel lines 
  
// Function to find the distance 
// between parallel lines
function dist($m, $b1, $b2
    $d = abs($b2 - $b1) / (($m * $m) - 1); 
    return $d
  
// Driver Code 
$m = 2;
$b1 = 4;
$b2 = 3; 
  
echo dist($m, $b1, $b2);
  
// This code is contributed by Ryuga
?>

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Output:

0.333333


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