Length of the transverse common tangent between the two non intersecting circles

Given two circles of given radii, having there centres a given distance apart, such that the circles don’t touch each other. The task is to find the length of the transverse common tangent between the circles.

Examples:

Input: r1 = 4, r2 = 6, d = 12
Output: 6.63325

Input: r1 = 7, r2 = 9, d = 21
Output: 13.6015



Approach:

  1. Let the radii of the circles be r1 & r2 respectively.
  2. Let the distance between the centers be d units.
  3. Draw a line O’R parallel to PQ,
  4. angle OPQ = angle RPQ = 90 deg
    angle O’QP = 90 deg
    { line joining the center of the circle to the point of contact makes an angle of 90 degree with the tangent }
  5. angle RPQ + angle O’QP = 180 deg
    PR || O’Q
  6. Since opposite sides are parallel and interior angles are 90, therefore O’PQR is a rectangle.
  7. O’Q = RP = r2 and PQ = O’R
  8. In triangle OO’R

    angle ORO’ = 90 deg
    By Pythagoras theorem,
    OR^2 + O’R^2 = OO’^2
    O’R^2 = OO’^2 – OR^2
    O’R^2 = d^2 – (r1+r2)^2
    O’R^2 = √(d^2 – (r1+r2)^2)

  Length of transverse common tangent = sqrt((distance between centers)^2 - (sum of radii)^2)

C++

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// C++ program to find the length
// of the transverse common tangent
// between two circles which
// do not touch each other
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the length
// of the transverse common tangent
void lengthOfTangent(double r1, double r2, double d)
{
  
    cout << "The length of the transverse"
         << " common tangent is "
         << sqrt(pow(d, 2) - pow((r1 + r2), 2))
         << endl;
}
  
// Driver code
int main()
{
    double r1 = 4, r2 = 6, d = 12;
    lengthOfTangent(r1, r2, d);
    return 0;
}

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Java

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// Java program to find the length
// of the transverse common tangent
// between two circles which
// do not touch each other
class GFG {
  
    // Function to find the length
    // of the transverse common tangent
    static void lengthOfTangent(double r1,
                                double r2, double d)
    {
  
        System.out.println("The length of the transverse"
                           + " common tangent is "
                           + Math.sqrt(Math.pow(d, 2)
                                       - Math.pow((r1 + r2), 2)));
    }
  
    // Driver code
    public static void main(String args[])
    {
        double r1 = 4, r2 = 6, d = 12;
        lengthOfTangent(r1, r2, d);
    }
}
  
// This code has been contributed by 29AjayKumar

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Python3

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# python 3 program to find the length
# of the transverse common tangent
# between two circles which
# do not touch each other
from math import sqrt, pow
  
# Function to find the length
# of the transverse common tangent
def lengthOfTangent(r1, r2, d):
    print("The length of the transverse",
                     "common tangent is"
          '{0:.6g}'.format(sqrt(pow(d, 2) - 
                                pow((r1 + r2), 2))))
  
# Driver code
if __name__ == '__main__':
    r1 = 4
    r2 = 6
    d = 12
    lengthOfTangent(r1, r2, d)
      
# This code is contributed by
# Surendra_Gangwar

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C#

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// C# program to find the length
// of the transverse common tangent
// between two circles which
// do not touch each other
using System;
  
class GFG {
    // Function to find the length
    // of the transverse common tangent
    static void lengthOfTangent(double r1,
                                double r2, double d)
    {
  
        Console.WriteLine("The length of the transverse"
                          + " common tangent is "
                          + Math.Sqrt(Math.Pow(d, 2)
                                      - Math.Pow((r1 + r2), 2)));
    }
  
    // Driver code
    static public void Main()
    {
        double r1 = 4, r2 = 6, d = 12;
        lengthOfTangent(r1, r2, d);
    }
}
  
// This code has been contributed by ajit.

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PHP

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<?php
// PHP program to find the length 
// of the transverse common tangent 
// between two circles which 
// do not touch each other 
  
// Function to find the length 
// of the transverse common tangent 
function lengthOfTangent($r1, $r2, $d
  
    echo "The length of the transverse common tangent is ",
    sqrt(pow($d, 2) - pow(($r1 + $r2), 2)) ;
  
// Driver code 
$r1 = 4; $r2 = 6; $d = 12; 
lengthOfTangent($r1, $r2, $d); 
  
// This code is contributed by AnkitRai01
?>

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Output:

The length of the transverse common tangent is 6.63325


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