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 <bits/stdc++.h>#include <iomanip>#include <iostream>#include <math.h>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 Code if __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 |
PHP
<?php// PHP program to find equation of // an ellipse using focus and directrix. // Function to find equation of ellipse. function 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); $fixed; // echo setprecision(2); echo "Equation of ellipse is \n" , $a1, " x^2 + ", $b1 , " y^2 + ", $c1 , " x + " , $d1 , " y + ", $e1 , " xy + " , $f1 , " = 0"; } // Driver Code $x1 = 1; $y1 = 1; $a = 1;$b = -1;$c = 3;$e = 0.5 * 0.5; equation_ellipse($x1, $y1, $a, $b, $c, $e); // This code is contributed by jit_t?> |
Javascript
<script>// Javascript program to find equation // of an ellipse using focus and directrix.// Function to find equation of ellipse.function equation_ellipse(x1, y1, a, b, c, e){ var t = a * a + b * b; var a1 = t - e * (a * a); var b1 = t - e * (b * b); var c1 = (-2 * t * x1) - (2 * e * c * a); var d1 = (-2 * t * y1) - (2 * e * c * b); var e1 = -2 * e * a * b; var f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1); document.write("Equation of ellipse is " + "<br>"); document.write(a1+" x^2 + "+ b1 + " y^2 + "+ c1 + " x + " + d1 + " y + " + e1 + " xy + " + f1 + " = 0");}// Driver Code var x1 = 1, y1 = 1, a = 1, b = -1, c = 3, e = 0.5 * 0.5;equation_ellipse(x1, y1, a, b, c, e);// This code is contributed by Khushboogoyal499 </script> |
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
Time Complexity: O(1)
Auxiliary Space: O(1)
Feeling lost in the world of random DSA topics, wasting time without progress? It’s time for a change! Join our DSA course, where we’ll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
