# Equation of parabola from its focus and directrix

• Last Updated : 20 Sep, 2021

We are given focus(x, y) and directrix(ax + by + c) of a parabola and we have to find the equation of parabola using its focus and directrix.
Examples :

Input: x1 = 0, y1 = 0, a = 2, b = 1, c = 2
Output: equation of parabola is 16.0 x^2 + 9.0 y^2 + -12.0 x + 16.0 y + 24.0 xy + -4.0 = 0.
Input: x1 = -1, y1 = -2, a = 1, b = -2, c = 3
Output:equation of parabola is 4.0 x^2 + 1.0 y^2 + 4.0 x + 32.0 y + 4.0 xy + 16.0 = 0.

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In case you wish to attend live classes with experts, please refer DSA Live Classes for Working Professionals and Competitive Programming Live for Students. Let P(x, y) be any point on the parabola whose focus S(x1, y1) and the directrix is the straight line ax + by + c =0.
Draw PM perpendicular from P on the directrix. then by definition pf parabola distance SP = PM
SP^2 = PM^2

`(x - x1)^2 + (y - y1)^2 = ( ( 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  = ( ( 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  = ( ( 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  = a^2*x^2 + b^2*y^2 + 2*a*x*b*y + c^2 + 2*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  = a^2*x^2 + b^2*y^2 + 2*a*x*b*y + c^2 + 2*c*a*x + 2*c*b*y
t*x^2 - a^2*x^2 +  t*y^2 - b^2*y^2 - 2*t*x1*x - 2*c*a*x - 2*t*y1*y - 2*c*b*y - 2*a*x*b*y - c^2  + t*x1^2 + t*y1^2 =0.```

This can be compared with 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 :

## C++

 `// C++ program to find equation of a parbola``// using focus and directrix.``#include ``#include ``#include ``#include ` `using` `namespace` `std;` `// Function to find equation of parabola.``void` `equation_parabola(``float` `x1, ``float` `y1,``                       ``float` `a, ``float` `b, ``float` `c)``{``    ``float` `t = a * a + b * b;``    ``float` `a1 = t - (a * a);``    ``float` `b1 = t - (b * b);``    ``float` `c1 = (-2 * t * x1) - (2 * c * a);``    ``float` `d1 = (-2 * t * y1) - (2 * c * b);``    ``float` `e1 = -2 * a * b;``    ``float` `f1 = (-c * c) + (t * x1 * x1) + (t * y1 * y1);``    ``std::cout << std::fixed;``    ``std::cout << std::setprecision(1);``    ``cout << ``"equation of parabola is "` `<< a1``         ``<< ``" x^2 + "` `<< b1 << ``" y^2 + "``         ``<< c1 << ``" x + "` `<< d1 << ``" y + "``         ``<< e1 << ``" xy + "` `<< f1 << ``" = 0."``;``}` `// Driver Code``int` `main()``{``    ``float` `x1 = 0;``    ``float` `y1 = 0;``    ``float` `a = 3;``    ``float` `b = -4;``    ``float` `c = 2;``    ``equation_parabola(x1, y1, a, b, c);``    ``return` `0;``}``// This code is contributed by Amber_Saxena.`

## Java

 `// Java program to find equation of a parbola``// using focus and directrix.``import` `java.util.*;` `class` `solution``{` `//Function to find equation of parabola.``static` `void` `equation_parabola(``float` `x1, ``float` `y1,``                    ``float` `a, ``float` `b, ``float` `c)``{``    ``float` `t = a * a + b * b;``    ``float` `a1 = t - (a * a);``    ``float` `b1 = t - (b * b);``    ``float` `c1 = (-``2` `* t * x1) - (``2` `* c * a);``    ``float` `d1 = (-``2` `* t * y1) - (``2` `* c * b);``    ``float` `e1 = -``2` `* a * b;``    ``float` `f1 = (-c * c) + (t * x1 * x1) + (t * y1 * y1);``    ``System.out.println( ``"equation of parabola is "``+ a1+``                        ``" x^2 + "` `+b1 +``" y^2 + "``+``                        ``c1 + ``" x + "` `+d1 + ``" y + "``                        ``+ e1+``" xy + "` `+ f1 +``" = 0."``);` `}` `// Driver Code``public` `static` `void` `main(String arr[])``{``    ``float` `x1 = ``0``;``    ``float` `y1 = ``0``;``    ``float` `a = ``3``;``    ``float` `b = -``4``;``    ``float` `c = ``2``;``    ``equation_parabola(x1, y1, a, b, c);` `}` `}`

## Python3

 `# Python3 program to find equation of a parbola``# using focus and directrix.` `# Function to find equation of parabola.``def` `equation_parabola(x1, y1, a, b, c) :`` ` `    ``t ``=` `a ``*` `a ``+` `b ``*` `b``    ``a1 ``=` `t ``-` `(a ``*` `a)``    ``b1 ``=` `t ``-` `(b ``*` `b);``    ``c1 ``=` `(``-``2` `*` `t ``*` `x1) ``-` `(``2` `*` `c ``*` `a)``    ``d1 ``=` `(``-``2` `*` `t ``*` `y1) ``-` `(``2` `*` `c ``*` `b)``    ``e1 ``=` `-``2` `*` `a ``*` `b``    ``f1 ``=` `(``-``c ``*` `c) ``+` `(t ``*` `x1 ``*` `x1) ``+` `(t ``*` `y1 ``*` `y1)``    ``print``(``"equation of parabola 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 ``=` `0``,``0``,``3``,``-``4``,``2``    ``equation_parabola(x1, y1, a, b, c)` `# This code is contributed by Ryuga`

## C#

 `// C# program to find equation of a parbola``// using focus and directrix.``using` `System;` `class` `solution``{` `//Function to find equation of parabola.``static` `void` `equation_parabola(``float` `x1, ``float` `y1,``                    ``float` `a, ``float` `b, ``float` `c)``{``    ``float` `t = a * a + b * b;``    ``float` `a1 = t - (a * a);``    ``float` `b1 = t - (b * b);``    ``float` `c1 = (-2 * t * x1) - (2 * c * a);``    ``float` `d1 = (-2 * t * y1) - (2 * c * b);``    ``float` `e1 = -2 * a * b;``    ``float` `f1 = (-c * c) + (t * x1 * x1) + (t * y1 * y1);``    ``Console.WriteLine( ``"equation of parabola is "``+ a1+``                        ``" x^2 + "` `+b1 +``" y^2 + "``+``                        ``c1 + ``" x + "` `+d1 + ``" y + "``                        ``+ e1+``" xy + "` `+ f1 +``" = 0."``);` `}` `// Driver Code``public` `static` `void` `Main()``{``    ``float` `x1 = 0;``    ``float` `y1 = 0;``    ``float` `a = 3;``    ``float` `b = -4;``    ``float` `c = 2;``    ``equation_parabola(x1, y1, a, b, c);` `// This Code is contributed``// by shs``}` `}`

## Javascript

 ``
Output:
`equation of parabola is 16.0 x^2 + 9.0 y^2 + -12.0 x + 16.0 y + 24.0 xy + -4.0 = 0.`

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