Length of direct common tangent between two intersecting Circles

Given two circles, of given radii, have their centres a given distance apart, such that the circles intersect each other at two points. The task is to find the length of the direct common tangent between the circles.
Examples:

Input: r1 = 4, r2 = 6, d = 3  
Output: 2.23607

Input: r1 = 14, r2 = 43, d = 35
Output: 19.5959

Approach:

• Let the radii of the circles be r1 & r2 respectively.

• Let the distance between the centers be d units.

• Draw a line OR parallel to PQ

• angle OPQ = 90 deg 
  angle O’QP = 90 deg
  { line joining the centre of the circle to the point of contact makes an angle of 90 degrees with the tangent }

• angle OPQ + angle O’QP = 180 deg 
  OP || QR

• Since opposite sides are parallel and interior angles are 90, therefore OPQR is a rectangle.

• So OP = QR = r1 and PQ = OR = d

• In triangle OO’R
  angle ORO’ = 90
  By Pythagoras theorem
  OR^2 + O’R^2 = (OO’^2) 
  OR^2 + (r1-r2)^2 = d^2

• so, OR^2= d^2-(r1-r2)^2 
  OR = √{d^2-(r1-r2)^2}
  
  

Below is the implementation of the above approach:

C++

// C++ program to find
// the length of the direct
// common tangent between two circles
// which intersect each other
#include <bits/stdc++.h>
using namespace std;

// Function to find the length of the direct common tangent
void lengtang(double r1, double r2, double d)
{
    cout << "The length of the direct"
         <<" common tangent is "
         << sqrt(pow(d, 2) - pow((r1 - r2), 2))
         << endl;
}

// Driver code
int main()
{
    double r1 = 4, r2 = 6, d = 3;
    lengtang(r1, r2, d);
    return 0;
}

Java

// Java program to find
// the length of the direct
// common tangent between two circles
// which intersect each other
class GFG
{

    // Function to find the length of
    // the direct common tangent
    static void lengtang(double r1, double r2, double d)
    {
        System.out.println("The length of the direct"
                + " 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 = 3;
        lengtang(r1, r2, d);
    }
}

/* This code contributed by PrinciRaj1992 */

Python3

# Python program to find
# the length of the direct
# common tangent between two circles
# which intersect each other

# Function to find the length of
# the direct common tangent
def lengtang(r1, r2, d):
    print("The length of the direct common tangent is "
        ,((d** 2) - ((r1 - r2)** 2))**(1/2));


# Driver code
r1 = 4; r2 = 6; d = 3;
lengtang(r1, r2, d);

# This code has been contributed by 29AjayKumar

C#

// C# program to find
// the length of the direct
// common tangent between two circles
// which intersect each other
using System;

class GFG
{

    // Function to find the length of
    // the direct common tangent
    static void lengtang(double r1, double r2, double d)
    {
        Console.WriteLine("The length of the direct"
                + " 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 = 3;
        lengtang(r1, r2, d);
    }
}

/* This code contributed by PrinciRaj1992 */

PHP

<?php
// PHP program to find
// the length of the direct
// common tangent between two circles
// which intersect each other

// Function to find the length of
// the direct common tangent
function lengtang($r1, $r2, $d)
{
    echo "The length of the direct common tangent is "
        ,sqrt(pow($d, 2) - pow(($r1 - $r2), 2)) ;
}

// Driver code
$r1 = 4; $r2 = 6; $d = 3;
lengtang($r1, $r2, $d);

// This code is contributed by AnkitRai01
?>

Javascript

<script>

// javascript program to find
// the length of the direct
// common tangent between two circles
// which intersect each other

// Function to find the length of
// the direct common tangent
function lengtang(r1 , r2 , d)
{
    document.write("The length of the direct"
            + " common tangent is "
            + (Math.sqrt(Math.pow(d, 2) -
                Math.pow((r1 - r2), 2))).toFixed(5));
}

// Driver code
var r1 = 4, r2 = 6, d = 3;
lengtang(r1, r2, d);


// This code contributed by Princi Singh

</script>

Output:

The length of the direct common tangent is 2.23607

Time Complexity: O(logn) because using inbuilt sqrt and pow function

Auxiliary Space: O(1)