# 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 ```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution. 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 ` `#include ` `#include ` `#include ` ` `  `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

 ` `

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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