We are given focus(x, y) and directrix(ax + by + c) of a parabola and we have to find the equation of the 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 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 :
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 Codeint 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
<script>// javascript program to find equation of a parbola // using focus and directrix. // Function to find equation of parabola. function equation_parabola(x1 , y1 , a , b , c) { var t = a * a + b * b; var a1 = t - (a * a); var b1 = t - (b * b); var c1 = (-2 * t * x1) - (2 * c * a); var d1 = (-2 * t * y1) - (2 * c * b); var e1 = -2 * a * b; var f1 = (-c * c) + (t * x1 * x1) + (t * y1 * y1); document.write("equation of parabola is " + a1 + " x^2 + " + b1 + " y^2 + " + c1 + " x + " + d1 + " y + " + e1 + " xy + " + f1 + " = 0."); } // Driver Code var x1 = 0; var y1 = 0; var a = 3; var b = -4; var c = 2; equation_parabola(x1, y1, a, b, c);// This code contributed by gauravrajput1</script> |
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.
Time Complexity: O(1)
Auxiliary Space: O(1)
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