Given two circles, of given radii, that touch each other externally. The task is to find the length of the direct common tangent between the circles.
Examples:
Input: r1 = 5, r2 = 9 Output: 13.4164 Input: r1 = 11, r2 = 13 Output: 23.9165
Approach
- Let the radii be r1 & r2 respectively.
- 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
OP || QR - Since opposite sides are parallel and interior angles are 90, therefore OPQR is a rectangle.
- So OP = QR = r1 and PQ = OR = r1+r2
- In triangle OO’R
angle ORO’ = 90
By Pythagoras theorem
OR^2 + O’R^2 = OO’^2
OO’^2 = (r1+r2)^2 + (r1-r2)^2 - So, OO’ = 2√(r1*r2)
Below is the implementation of the above approach:
C++
// C++ program to find the length of the direct// common tangent between two circles// which externally touch each other#include <bits/stdc++.h>using namespace std;// Function to find the length// of the direct common tangentvoid lengtang(double r1, double r2){ cout << "The length of the " << "direct common tangent is " << 2 * sqrt(r1 * r2) << endl;}// Driver codeint main(){ double r1 = 5, r2 = 9; lengtang(r1, r2); return 0;} |
Java
// Java program to find the length of the direct // common tangent between two circles // which externally touch each other class GFG { // Function to find the length // of the direct common tangent static void lengtang(double r1, double r2) { System.out.println("The length of the " + "direct common tangent is " + (2 * Math.sqrt(r1 * r2))); } // Driver code public static void main(String[] args) { double r1 = 5, r2 = 9; lengtang(r1, r2); }}// This code contributed by Rajput-Ji |
Python3
# Python3 program to find the length # of the direct common tangent # between two circles which # externally touch each other # Function to find the length # of the direct common tangent def lengtang(r1, r2): print("The length of the direct", "common tangent is", 2 * (r1 * r2)**(1 / 2)); # Driver code r1 = 5; r2 = 9; lengtang(r1, r2); # This code contributed # by PrinciRaj1992 |
C#
// C# program to find the length of the direct // common tangent between two circles // which externally touch each otherusing System;class GFG{ // Function to find the length // of the direct common tangent static void lengtang(double r1, double r2) { Console.WriteLine("The length of the " + "direct common tangent is " + (2 * Math.Sqrt(r1 * r2))); } // Driver code static public void Main () { double r1 = 5, r2 = 9; lengtang(r1, r2); }}// This code contributed by ajit. |
PHP
<?php// PHP program to find the length of the direct // common tangent between two circles // which externally touch each other // Function to find the length // of the direct common tangent function lengtang($r1, $r2) { echo "The length of the " , "direct common tangent is " , 2 * sqrt($r1 * $r2) ; } // Driver code $r1 = 5; $r2 = 9; lengtang($r1, $r2); // This code is contributed by AnkitRai01?> |
Javascript
<script>// javascript program to find the length of the direct // common tangent between two circles // which externally touch each other // Function to find the length // of the direct common tangent function lengtang(r1 , r2) { document.write("The length of the " + "direct common tangent is " + (2 * Math.sqrt(r1 * r2)).toFixed(5));}// Driver code var r1 = 5, r2 = 9;lengtang(r1, r2);// This code contributed by Princi Singh </script> |
Output:
The length of the direct common tangent is 13.4164
Time Complexity: O(log(n)) because using inbuilt sqrt function
Auxiliary Space: O(1)
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