椭圆中可内接的最大矩形的面积

给定一个椭圆，长轴长度为2a和2b 。任务是找到可以内接的最大矩形的面积。
例子：

Input: a = 4, b = 3
Output: 24

Input: a = 10, b = 8
Output: 160

方法：
让矩形的右上角有坐标(x, y) ，
然后是矩形的面积， A = 4*x*y 。
现在，

Equation of ellipse, (x²/a²) + (y²/b²) = 1
Thinking of the area as a function of x, we have
dA/dx = 4xdy/dx + 4y
Differentiating equation of ellipse with respect to x, we have
2x/a² + (2y/b²)dy/dx = 0,
so,
dy/dx = -b²x/a²y,
and
dAdx = 4y – (4b²x²/a²y)
Setting this to 0 and simplifying, we have y² = b²x²/a².
From equation of ellipse we know that,
y²=b² – b²x²/a2
Thus, y²=b² – y², 2y²=b², and y²b² = 1/2.
Clearly, then, x²a² = 1/2 as well, and the area is maximized when
x= a/√2 and y=b/√2
So the maximum area Area, A_max = 2ab

编程需要懂一点英语

下面是上述方法的实现：

C++

// C++ Program to find the biggest rectangle
// which can be inscribed within the ellipse
#include 
using namespace std;
 
// Function to find the area
// of the rectangle
float rectanglearea(float a, float b)
{
 
    // a and b cannot be negative
    if (a < 0 || b < 0)
        return -1;
 
    // area of the rectangle
    return 2 * a * b;
}
 
// Driver code
int main()
{
    float a = 10, b = 8;
    cout << rectanglearea(a, b) << endl;
    return 0;
}

Java

// Java Program to find the biggest rectangle
// which can be inscribed within the ellipse
 
import java.util.*;
import java.lang.*;
import java.io.*;
 
class GFG{
// Function to find the area
// of the rectangle
static float rectanglearea(float a, float b)
{
  
    // a and b cannot be negative
    if (a < 0 || b < 0)
        return -1;
  
    // area of the rectangle
    return 2 * a * b;
}
  
// Driver code
public static void main(String args[])
{
    float a = 10, b = 8;
    System.out.println(rectanglearea(a, b));
}
}

Python 3

# Python 3 Program to find the biggest rectangle
# which can be inscribed within the ellipse
 
#  Function to find the area
# of the rectangle
def rectanglearea(a, b) :
 
    # a and b cannot be negative
    if a < 0 or b < 0 :
        return -1
 
    # area of the rectangle
    return 2 * a * b
  
 
# Driver code    
if __name__ == "__main__" :
 
    a, b = 10, 8
    print(rectanglearea(a, b))
 
 
# This code is contributed by ANKITRAI1

C#

// C# Program to find the
// biggest rectangle which
// can be inscribed within
// the ellipse
using System;
 
class GFG
{
// Function to find the area
// of the rectangle
static float rectanglearea(float a,
                           float b)
{
 
    // a and b cannot be negative
    if (a < 0 || b < 0)
        return -1;
 
    // area of the rectangle
    return 2 * a * b;
}
 
// Driver code
public static void Main()
{
    float a = 10, b = 8;
    Console.WriteLine(rectanglearea(a, b));
}
}
 
// This code is contributed
// by inder_verma

PHP

Javascript

输出：