Equation of ellipse from its focus, directrix, and eccentricity

• Last Updated : 09 Apr, 2021

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 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 Codeint 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 Codepublic 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 Codeif __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 Codepublic 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



Javascript


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

My Personal Notes arrow_drop_up