# 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 donâ€™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 = 12Output:6.63325Input:r1 = 7, r2 = 9, d = 21Output:13.6015

**Approach**:

- Let the radii of the circles be
**r1**&**r2**respectively. - Let the distance between the centers be
**d**units. - Draw a line O’R parallel to PQ,
**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 }

**angle RPQ + angle O’QP = 180 deg****PR || O’Q**- Since opposite sides are parallel and interior angles are 90, therefore
**O’PQR**is a rectangle. **O’Q = RP = r2 and PQ = O’R**- 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 = âˆš(d^2 – (r1+r2)^2)**

## C++

`// 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;` `}` |

## Java

`// 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` |

## Python3

`# 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` |

## C#

`// 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.` |

## PHP

`<?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` `?>` |

## Javascript

`<script>` `// javascript 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)` `{` ` ` `document.write(` `"The length of the transverse"` ` ` `+ ` `" common tangent is "` ` ` `+ Math.sqrt(Math.pow(d, 2)` ` ` `- Math.pow((r1 + r2), 2)));` `}` `// Driver code` `var` `r1 = 4, r2 = 6, d = 12;` `lengthOfTangent(r1, r2, d);` `// This code contributed by Princi Singh ` `</script>` |

**Output:**

The length of the transverse common tangent is 6.63325

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

**Auxiliary Space:** O(1)