Ratio of the distance between the centers of the circles and the point of intersection of two direct common tangents to the circles
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++
#include <bits/stdc++.h>
using namespace std;
int GCD( int a, int b)
{
return (b != 0 ? GCD(b, a % b) : a);
}
void ratiotang( int r1, int r2)
{
cout << "The ratio is "
<< r1 / GCD(r1, r2)
<< " : "
<< r2 / GCD(r1, r2)
<< endl;
}
int main()
{
int r1 = 4, r2 = 6;
ratiotang(r1, r2);
return 0;
}
|
Java
class GFG {
static int GCD( int a, int b)
{
return (b != 0 ? GCD(b, a % b) : a);
}
static void ratiotang( int r1, int r2)
{
System.out.println( "The ratio is "
+ r1 / GCD(r1, r2)
+ " : "
+ r2 / GCD(r1, r2));
}
public static void main(String args[])
{
int r1 = 4 , r2 = 6 ;
ratiotang(r1, r2);
}
}
|
Python3
from math import gcd
def ratiotang(r1, r2):
print ( "The ratio is" , int (r1 / gcd(r1, r2)), ":" ,
int (r2 / gcd(r1, r2)))
if __name__ = = '__main__' :
r1 = 4
r2 = 6
ratiotang(r1, r2)
|
C#
using System;
class GFG
{
static int GCD( int a, int b)
{
return (b != 0 ? GCD(b, a % b) : a);
}
static void ratiotang( int r1, int r2)
{
Console.WriteLine( "The ratio is "
+ r1 / GCD(r1, r2)
+ " : "
+ r2 / GCD(r1, r2));
}
public static void Main(String[] args)
{
int r1 = 4, r2 = 6;
ratiotang(r1, r2);
}
}
|
PHP
<?php
function GCD( $a , $b )
{
return ( $b != 0 ? GCD( $b , $a % $b ) : $a );
}
function ratiotang( $r1 , $r2 )
{
echo "The ratio is " , $r1 / GCD( $r1 , $r2 ),
" : " , $r2 / GCD( $r1 , $r2 ) ;
}
$r1 = 4; $r2 = 6;
ratiotang( $r1 , $r2 );
?>
|
Javascript
<script>
function GCD(a , b)
{
return (b != 0 ? GCD(b, a % b) : a);
}
function ratiotang(r1 , r2)
{
document.write( "The ratio is "
+ r1 / GCD(r1, r2)
+ " : "
+ r2 / GCD(r1, r2));
}
var r1 = 4, r2 = 6;
ratiotang(r1, r2);
</script>
|
Output:
The ratio is 2 : 3
Time Complexity: O(log(min(a, b)))
Auxiliary Space: O(log(min(a, b)))
Last Updated :
07 Jun, 2022
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