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 tangentvoid 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 codeint 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 otherclass 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 coder1 = 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 otherusing 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 tangentfunction 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 codevar 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)
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