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Equation of parabola from its focus and directrix
  • Last Updated : 20 Dec, 2018

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.



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 <bits/stdc++.h>
#include <iomanip>
#include <iostream>
#include <math.h>
  
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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