Find the Area of the Largest Rectangle That Can Be Inscribed in the Ellipse X2 A2 + Y2 B2 = 1
Surface area of Largest rectangle that tin can be inscribed in an Ellipse
Given an ellipse, with major axis length 2a & 2b. The task is to find the area of the largest rectangle that can exist inscribed in information technology.
Examples:
Input: a = 4, b = three Output: 24 Input: a = 10, b = viii Output: 160
Approach:
Let the upper right corner of the rectangle has co-ordinates (x, y),
And then the area of rectangle, A = iv*x*y.
Now,
Equation of ellipse, (x2/atwo) + (y2/b2) = 1
Thinking of the area as a office of x, we have
dA/dx = 4xdy/dx + 4y
Differentiating equation of ellipse with respect to x, we have
2x/aii + (2y/b2)dy/dx = 0,
so,
dy/dx = -bii10/a2y,
and
dAdx = 4y – (4b2x2/a2y)
Setting this to 0 and simplifying, nosotros have y2 = biix2/a2 .
From equation of ellipse nosotros know that,
y2=btwo – b2ten2/a2
Thus, y2=b2 – yii , 2y2=b2 , and y2btwo = ane/ii.
Clearly, and then, 102a2 = 1/ii also, and the expanse is maximized when
ten= a/√2 and y=b/√2
And then the maximum expanse Expanse, Amax = 2ab
Beneath is the implementation of the above approach:
C++
#include <bits/stdc++.h>
using namespace std;
bladder rectanglearea( bladder a, float b)
{
if (a < 0 || b < 0)
return -1;
return 2 * a * b;
}
int master()
{
float a = 10, b = eight;
cout << rectanglearea(a, b) << endl;
return 0;
}
Java
import java.util.*;
import java.lang.*;
import java.io.*;
class GFG{
static float rectanglearea( float a, float b)
{
if (a < 0 || b < 0 )
return - ane ;
return ii * a * b;
}
public static void chief(String args[])
{
float a = 10 , b = 8 ;
System.out.println(rectanglearea(a, b));
}
}
Python 3
def rectanglearea(a, b) :
if a < 0 or b < 0 :
return - i
return 2 * a * b
if __name__ = = "__main__" :
a, b = 10 , 8
print (rectanglearea(a, b))
C#
using System;
class GFG
{
static float rectanglearea( bladder a,
bladder b)
{
if (a < 0 || b < 0)
render -1;
render 2 * a * b;
}
public static void Main()
{
float a = x, b = 8;
Panel.WriteLine(rectanglearea(a, b));
}
}
PHP
<?php
role rectanglearea( $a , $b )
{
if ( $a < 0 or $b < 0)
return -one;
return 2 * $a * $b ;
}
$a = 10; $b = viii;
echo rectanglearea( $a , $b );
?>
Javascript
<script>
function rectanglearea(a , b)
{
if (a < 0 || b < 0)
return -ane;
render 2 * a * b;
}
var a = ten, b = eight;
document.write(rectanglearea(a, b));
</script>
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Source: https://www.geeksforgeeks.org/area-of-largest-rectangle-that-can-be-inscribed-in-an-ellipse/
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