Open In App

# 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 ``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

 ``

## Javascript

 ``

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)