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/a^two) + (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=b^two – b²ten²/a2
Thus, y²=b² – yⁱⁱ , 2y²=b² , and y²b^two = 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, A_max = 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/