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Equation of ellipse from its focus, directrix, and eccentricity
  • Last Updated : 09 Apr, 2021

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++




// 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;
}

Java




// 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

Python3




# 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

C#




// 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

PHP




<?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
?>

Javascript




<script>
 
// Javascript 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)
{
    var t = a * a + b * b;
    var a1 = t - e * (a * a);
    var b1 = t - e * (b * b);
    var c1 = (-2 * t * x1) - (2 * e * c * a);
    var d1 = (-2 * t * y1) - (2 * e * c * b);
    var e1 = -2 * e * a * b;
    var f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1);
 
    document.write("Equation of ellipse is " + "<br>");
    document.write(a1+" x^2 + "+ b1 + " y^2 + "+ c1 + " x + "
                    + d1 + " y + " + e1 + " xy + " + f1 + " = 0");
}
 
// Driver Code
var 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 Khushboogoyal499
     
</script>
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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