# Equation of parabola from its focus and directrix

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.

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

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 ` `}  ` ` `  `}  `

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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