Length of the direct common tangent between two externally touching circles
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++
#include <bits/stdc++.h>
using namespace std;
void lengtang( double r1, double r2)
{
cout << "The length of the "
<< "direct common tangent is "
<< 2 * sqrt (r1 * r2) << endl;
}
int main()
{
double r1 = 5, r2 = 9;
lengtang(r1, r2);
return 0;
}
|
Java
class GFG
{
static void lengtang( double r1, double r2)
{
System.out.println( "The length of the "
+ "direct common tangent is "
+ ( 2 * Math.sqrt(r1 * r2)));
}
public static void main(String[] args)
{
double r1 = 5 , r2 = 9 ;
lengtang(r1, r2);
}
}
|
Python3
def lengtang(r1, r2):
print ( "The length of the direct" ,
"common tangent is" ,
2 * (r1 * r2) * * ( 1 / 2 ));
r1 = 5 ; r2 = 9 ;
lengtang(r1, r2);
|
C#
using System;
class GFG
{
static void lengtang( double r1, double r2)
{
Console.WriteLine( "The length of the "
+ "direct common tangent is "
+ (2 * Math.Sqrt(r1 * r2)));
}
static public void Main ()
{
double r1 = 5, r2 = 9;
lengtang(r1, r2);
}
}
|
PHP
<?php
function lengtang( $r1 , $r2 )
{
echo "The length of the "
, "direct common tangent is "
, 2 * sqrt( $r1 * $r2 ) ;
}
$r1 = 5; $r2 = 9;
lengtang( $r1 , $r2 );
?>
|
Javascript
<script>
function lengtang(r1 , r2)
{
document.write( "The length of the "
+ "direct common tangent is "
+ (2 * Math.sqrt(r1 * r2)).toFixed(5));
}
var r1 = 5, r2 = 9;
lengtang(r1, r2);
</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)
Last Updated :
03 Aug, 2022
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