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 tangent void lengtang( double r1, double r2)
{ cout << "The length of the "
<< "direct common tangent is "
<< 2 * sqrt (r1 * r2) << endl;
} // Driver code int 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 other using 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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