Equation of ellipse from its focus, directrix, and eccentricity

Given focus(x, y), directrix(ax + by + c) and eccentricity e of an ellipse, the task is to find the equation of ellipse using its focus, directrix, and eccentricity.

Examples:

Input: x1 = 1, y1 = 1, a = 1, b = -1, c = 3, e = 0.5
Output: 1.75 x^2 + 1.75 y^2 + -5.50 x + -2.50 y + 0.50 xy + 1.75 = 0

Input: x1 = -1, y1 = 1, a = 1, b = -1, c = 3, e = 0.5
Output: 1.75 x^2 + 1.75 y^2 + 2.50 x + -2.50 y + 0.50 xy + 1.75 = 0 



Let P(x, y) be any point on the ellipse whose focus S(x1, y1), directrix is the straight line ax + by + c = 0 and eccentricity is e.
Draw PM perpendicular from P on the directrix. Then by definition of ellipse distance SP = e * PM => SP^2 = (e * PM)^2

(x – x1)^2 + (y – y1)^2 = e * ( ( a*x + b*y + c ) / (sqrt( a*a + b*b )) ) ^ 2

let ( a*a + b*b ) = t

x^2 + x1^2 – 2*x1*x + y^2 + y1^2 – 2*y1*y = e * ( ( a*x + b*y + c ) ^ 2 )/ t

on cross multiplying above we get

t*x^2 + t*x1^2 – 2*t*x1*x + t*y^2 + t*y1^2 – 2*t*y1*y = e * ( ( a*x + b*y + c ) ^ 2 )

t*x^2 + t*x1^2 – 2*t*x1*x + t*y^2 + t*y1^2 – 2*t*y1*y = e*a^2*x^2 + e*b^2*y^2 + 2*e*a*x*b*y + e*c^2 + 2*e*c*(a*x + b*y)

t*x^2 + t*x1^2 – 2*t*x1*x + t*y^2 + t*y1^2 – 2*t*y1*y = e*a^2*x^2 + e*b^2*y^2 + 2*e*a*x*b*y + e*c^2 + 2*e*c*a*x + 2*e*c*b*y

t*x^2 – e*a^2*x^2 + t*y^2 – e*b^2*y^2 – 2*t*x1*x – 2*e*c*a*x – 2*t*y1*y – 2*e*c*b*y – 2*e*a*x*b*y – e*c^2 + t*x1^2 + t*y1^2 =0

This can be compared with a general form that is:

a*x^2 + 2*h*x*y + b*y^2 + 2*g*x + 2*f*y + c = 0

Below is the implementation of the above approach:

C++

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// C++ program to find equation of an ellipse
// using focus and directrix.
#include <bits/stdc++.h>
#include <iomanip>
#include <iostream>
#include <math.h>
  
using namespace std;
  
// Function to find equation of ellipse.
void equation_ellipse(float x1, float y1,
                      float a, float b,
                      float c, float e)
{
    float t = a * a + b * b;
    float a1 = t - e * (a * a);
    float b1 = t - e * (b * b);
    float c1 = (-2 * t * x1) - (2 * e * c * a);
    float d1 = (-2 * t * y1) - (2 * e * c * b);
    float e1 = -2 * e * a * b;
    float f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1);
  
    cout << fixed;
    cout << setprecision(2);
    cout << "Equation of ellipse is \n"
         << a1
         << " x^2 + " << b1 << " y^2 + "
         << c1 << " x + " << d1 << " y + "
         << e1 << " xy + " << f1 << " = 0";
}
  
// Driver Code
int main()
{
    float x1 = 1, y1 = 1, a = 1, b = -1, c = 3, e = 0.5 * 0.5;
    equation_ellipse(x1, y1, a, b, c, e);
  
    return 0;
}

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Java

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// Java program to find equation of an ellipse
// using focus and directrix.
import java.util.*;
  
class solution
{
  
// Function to find equation of ellipse.
static void equation_ellipse(float x1, float y1,
                    float a, float b,
                    float c, float e)
{
    float t = a * a + b * b;
    float a1 = t - e * (a * a);
    float b1 = t - e * (b * b);
    float c1 = (-2 * t * x1) - (2 * e * c * a);
    float d1 = (-2 * t * y1) - (2 * e * c * b);
    float e1 = -2 * e * a * b;
    float f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1);
  
    System.out.println("Equation of ellipse is ");
    System.out.print(a1+" x^2 + "+ b1 + " y^2 + "+ c1 + " x + "
                    + d1 + " y + " + e1 + " xy + " + f1 + " = 0");
          
}
  
// Driver Code
public static void main(String arr[])
{
    float x1 = 1, y1 = 1, a = 1, b = -1, c = 3, e = (float)0.5 * (float)0.5;
    equation_ellipse(x1, y1, a, b, c, e);
  
}
}
  
//This code is contributed by Surendra_Gaangwar

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Python3

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# Python3 program to find equation of an ellipse 
# using focus and directrix.
  
# Function to find equation of ellipse. 
def equation_ellipse(x1, y1, a, b, c,  e) :
      
    t = a * a + b * b
    a1 = t - e * (a * a) 
    b1 = t - e * (b * b) 
    c1 = (-2 * t * x1) - (2 * e * c * a)
    d1 = (-2 * t * y1) - (2 * e * c * b) 
    e1 = -2 * e * a * b
    f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1) 
  
    print("Equation of ellipse is",a1,"x^2 +", b1 ,"y^2 +",
    c1, "x +" ,d1 ,"y +", e1 ,"xy +" , f1 ,"= 0")
   
  
# Driver Code 
if __name__ == "__main__"
  
    x1, y1, a, b, c, e = 1, 1, 1, -1, 3, 0.5 * 0.5
      
    equation_ellipse(x1, y1, a, b, c, e) 
  
# This code is contributed by Ryuga

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C#

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// C# program to find equation of an ellipse
// using focus and directrix.
  
class solution
{
  
// Function to find equation of ellipse.
static void equation_ellipse(float x1, float y1,
                    float a, float b,
                    float c, float e)
{
    float t = a * a + b * b;
    float a1 = t - e * (a * a);
    float b1 = t - e * (b * b);
    float c1 = (-2 * t * x1) - (2 * e * c * a);
    float d1 = (-2 * t * y1) - (2 * e * c * b);
    float e1 = -2 * e * a * b;
    float f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1);
  
    System.Console.WriteLine("Equation of ellipse is ");
    System.Console.WriteLine(a1+" x^2 + "+ b1 + " y^2 + "+ c1 + " x + "
                    + d1 + " y + " + e1 + " xy + " + f1 + " = 0");
          
}
  
// Driver Code
public static void Main()
{
    float x1 = 1, y1 = 1, a = 1, b = -1, c = 3, e = (float)0.5 * (float)0.5;
    equation_ellipse(x1, y1, a, b, c, e);
  
}
}
  
//This code is contributed by mits

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PHP

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<?php
// PHP program to find equation of 
// an ellipse using focus and directrix. 
  
// Function to find equation of ellipse. 
function equation_ellipse($x1, $y1, $a
                            $b, $c, $e
    $t = ($a * $a) + ($b * $b); 
    $a1 = $t - $e * ($a * $a); 
    $b1 = $t - $e * ($b * $b); 
    $c1 = (-2 * $t * $x1) - 
           (2 * $e * $c * $a); 
    $d1 = (-2 * $t * $y1) - 
           (2 * $e * $c * $b); 
    $e1 = -2 * $e * $a * $b
    $f1 = (-$e * $c * $c) + 
          ($t * $x1 * $x1) + ($t * $y1 * $y1); 
  
    $fixed
      
    // echo setprecision(2); 
    echo "Equation of ellipse is \n"
          $a1, " x^2 + ", $b1 , " y^2 + ",
          $c1 , " x + " , $d1 , " y + ",
          $e1 , " xy + " , $f1 , " = 0"
  
// Driver Code 
$x1 = 1; $y1 = 1; 
$a = 1;
$b = -1;
$c = 3;
$e = 0.5 * 0.5; 
equation_ellipse($x1, $y1, $a,
                 $b, $c, $e); 
  
// This code is contributed by jit_t
?>

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Output:

Equation of ellipse is 
1.75 x^2 + 1.75 y^2 + -5.50 x + -2.50 y + 0.50 xy + 1.75 = 0


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