Given two circles of given radii, such that the circles don’t touch each other, the task is to find the ratio of the distance between the centers of the circles and the point of intersection of two direct common tangents to the circles.
Examples:
Input: r1 = 4, r2 = 6 Output: 2:3 Input: r1 = 22, r2 = 43 Output: 22:43
Approach:
- Let the radii of the circles be r1 & r2 and the centers be C1 & C2 respectively.
- Let P be the point of intersection of two direct common tangents to the circles, and A1 & A2 be the point of contact of the tangents with the circles.
- In triangle PC1A1 & triangle PC2A2,
angle C1A1P = angle C2A2P = 90 deg { line joining the center of the circle to the point of contact makes an angle of 90 degree with the tangent },
also, angle A1PC1 = angle A2PC2
so, angle A1C1P = angle A2C2P
as angles are same, triangles PC1A1 & PC2A2 are similar. - So, due to similarity of the triangles,
C1P/C2P = C1A1/C2A2 = r1/r2
The ratio of the distance between the centers of the circles and the point of intersection of two direct common tangents to the circles = radius of the first circle : radius of the second circle
Below is the implementation of the above approach:
C++
// C++ program to find the ratio of the distance// between the centers of the circles// and the point of intersection of// two direct common tangents to the circles// which do not touch each other#include <bits/stdc++.h>using namespace std;// Function to find the GCDint GCD(int a, int b){ return (b != 0 ? GCD(b, a % b) : a);}// Function to find the ratiovoid ratiotang(int r1, int r2){ cout << "The ratio is " << r1 / GCD(r1, r2) << " : " << r2 / GCD(r1, r2) << endl;}// Driver codeint main(){ int r1 = 4, r2 = 6; ratiotang(r1, r2); return 0;} |
Java
// Java program to find the ratio of the distance// between the centers of the circles// and the point of intersection of// two direct common tangents to the circles// which do not touch each otherclass GFG { // Function to find the GCD static int GCD(int a, int b) { return (b != 0 ? GCD(b, a % b) : a); } // Function to find the ratio static void ratiotang(int r1, int r2) { System.out.println("The ratio is " + r1 / GCD(r1, r2) + " : " + r2 / GCD(r1, r2)); } // Driver code public static void main(String args[]) { int r1 = 4, r2 = 6; ratiotang(r1, r2); }}// This code has been contributed by 29AjayKumar |
Python3
# Python 3 program to find the ratio # of the distance between the centers # of the circles and the point of intersection # of two direct common tangents to the circles# which do not touch each other# Function to find the GCDfrom math import gcd# Function to find the ratiodef ratiotang(r1, r2): print("The ratio is", int(r1 / gcd(r1, r2)), ":", int(r2 / gcd(r1, r2)))# Driver codeif __name__ == '__main__': r1 = 4 r2 = 6 ratiotang(r1, r2)# This code is contributed by# Surendra_Gangwar |
C#
// C# program to find the ratio of the distance// between the centers of the circles// and the point of intersection of// two direct common tangents to the circles// which do not touch each other using System; class GFG{ // Function to find the GCD static int GCD(int a, int b) { return (b != 0 ? GCD(b, a % b) : a); } // Function to find the ratio static void ratiotang(int r1, int r2) { Console.WriteLine("The ratio is " + r1 / GCD(r1, r2) + " : " + r2 / GCD(r1, r2)); } // Driver code public static void Main(String[] args) { int r1 = 4, r2 = 6; ratiotang(r1, r2); }}// This code contributed by Rajput-Ji |
PHP
<?php// PHP program to find the ratio of the distance // between the centers of the circles // and the point of intersection of // two direct common tangents to the circles // which do not touch each other // Function to find the GCD function GCD($a, $b) { return ($b != 0 ? GCD($b, $a % $b) : $a); } // Function to find the ratio function ratiotang($r1, $r2) { echo "The ratio is ", $r1 / GCD($r1, $r2), " : ", $r2 / GCD($r1, $r2) ;} // Driver code $r1 = 4; $r2 = 6; ratiotang($r1, $r2); // This code is contributed by AnkitRai01?> |
Javascript
<script>// javascript program to find the ratio of the distance// between the centers of the circles// and the point of intersection of// two direct common tangents to the circles// which do not touch each other// Function to find the GCDfunction GCD(a , b){ return (b != 0 ? GCD(b, a % b) : a);}// Function to find the ratiofunction ratiotang(r1 , r2){ document.write("The ratio is " + r1 / GCD(r1, r2) + " : " + r2 / GCD(r1, r2));}// Driver codevar r1 = 4, r2 = 6;ratiotang(r1, r2);// This code is contributed by Princi Singh </script> |
Output:
The ratio is 2 : 3
Time Complexity: O(log(min(a, b)))
Auxiliary Space: O(log(min(a, b)))
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