Finding the vertex, focus and directrix of a parabola

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Problem – Find the vertex, focus and directrix of a parabola when the coefficients of its equation are given.
A set of points on a plain surface that forms a curve such that any point on that curve is equidistant from the focus is a parabola.
Vertex of a parabola is the coordinate from which it takes the sharpest turn whereas a is the straight line used to generate the curve.

The standard form of a parabola equation is $y=ax^2+bx+c$ . Given the values of a, b and c; our task is to find the coordinates of vertex, focus and the equation of the directrix.
Example –

Input : 5 3 2
Output : Vertex:(-0.3, 1.55)
         Focus: (-0.3, 1.6)
         Directrix: y=-198
Consult the formula below for explanation.

This problem is a simple example of implementations of formulae. Given below are the required set of formulae which will help us tackle the problem.

For a parabola in the form Vertex: Focus: Directrix:

C++

#include <iostream>
using namespace std;
 
// Function to calculate Vertex, Focus and Directrix
void parabola(float a, float b, float c)
{
    cout << "Vertex: (" << (-b / (2 * a)) << ", "
         << (((4 * a * c) - (b * b)) / (4 * a))
         << ")" << endl;
    cout << "Focus: (" << (-b / (2 * a)) << ", "
         << (((4 * a * c) - (b * b) + 1) / (4 * a))
         << ")" << endl;
    cout << "Directrix: y="
         << c - ((b * b) + 1) * 4 * a << endl;
}
 
// Driver Function
int main()
{
    float a = 5, b = 3, c = 2;
    parabola(a, b, c);
    return 0;
}

Java

// Java program to find the vertex,
// focus and directrix of a parabola
 
class GFG {
     
    // Function to calculate Vertex, 
    // Focus and Directrix
    static void parabola(float a, 
                         float b, float c)
    {
         
        System.out.println("Vertex: (" +
                          (-b / (2 * a)) + ", " +
                          (((4 * a * c) - (b * b)) /
                          (4 * a)) + ")");
                     
        System.out.println("Focus: (" + 
                          (-b / (2 * a)) + ", "    + 
                          (((4 * a * c) - (b * b) + 1) /
                          (4 * a)) + ")");
             
        System.out.println("Directrix:" + " y=" +
                          (int)(c - ((b * b) + 1) * 
                          4 * a));
    }
 
    // Driver Function
    public static void main(String[] args)
    {
        float a = 5, b = 3, c = 2;
         
        // Function calling
        parabola(a, b, c);
    }
}
 
// This code is contributed by 
// Smitha Dinesh Semwal

Python 3

# Function to calculate Vertex, 
# Focus and Directrix
def parabola(a, b, c):
 
    print("Vertex: (" , (-b / (2 * a)),
        ", ", (((4 * a * c) - (b * b)) 
            / (4 * a)), ")", sep = "")
               
    print("Focus: (" , (-b / (2 * a)),
    ", ", (((4 * a * c) - (b * b) + 1)
            / (4 * a)), ")", sep = "")
                
    print("Directrix: y=", c - ((b * b)
                + 1) * 4 * a, sep = "")
 
# Driver Function
a = 5
b = 3
c = 2
parabola(a, b, c)
 
# This code is contributed by Smitha.

C#

// C# program to find the vertex,
// focus and directrix of a parabola
using System;
 
class GFG {
     
    // Function to calculate Vertex, 
    // Focus and Directrix
    static void parabola(float a, 
                         float b, float c)
    {
        Console.WriteLine("Vertex: (" +
                         (-b / (2 * a)) + ", " +
                         (((4 * a * c) - (b * b)) /
                         (4 * a)) + ")");
                     
        Console.WriteLine("Focus: (" +
                         (-b / (2 * a)) + ", " +
                         (((4 * a * c) - (b * b) + 1) /
                         (4 * a)) + ")");
                 
        Console.Write("Directrix:" + " y=" + 
                     (int)(c - ((b * b) + 1) * 4 * a));
    }
 
    // Driver Function
    public static void Main()
    {
        float a = 5, b = 3, c = 2;
         
        // Function calling
        parabola(a, b, c);
    }
}
 
// This code is contributed by nitin mittal

PHP

<?php
// PHP program to Find the vertex,
// focus and directrix of a parabola
 
// Function to calculate Vertex, 
// Focus and Directrix
function parabola($a, $b, $c)
{
     
    echo "Vertex: (" , (-$b / (2 * $a)) , ", ",
        (((4 * $a * $c) - ($b * $b)) / (4 * $a)),
                                      ")", "\n" ;
    echo "Focus: (" , (-$b / (2 * $a)) , ", ",
        (((4 * $a * $c) - ($b * $b) + 1) / (4 * $a))
                                        , ")"," \n" ;
    echo "Directrix: y=",
        $c - (($b * $b) + 1) * 4 * $a ;
}
 
    // Driver Code
    $a = 5; $b = 3; $c = 2;
    parabola($a, $b, $c);
     
// This code is contributed by vt_m.
?>

Javascript

<script>
 
// JavaScript program to find the vertex,
// focus and directrix of a parabola
 
    // Function to calculate Vertex, 
    // Focus and Directrix
    function parabola(a, b, c)
    {
           
        document.write("Vertex: (" +
                          (-b / (2 * a)) + ", " +
                          (((4 * a * c) - (b * b)) /
                          (4 * a)) + ")" + "<br/>");
                       
       document.write("Focus: (" + 
                          (-b / (2 * a)) + ", "    + 
                          (((4 * a * c) - (b * b) + 1) /
                          (4 * a)) + ")" + "<br/>");
               
        document.write("Directrix:" + " y=" +
                          (c - ((b * b) + 1) * 
                          4 * a) + "<br/>");
    }
 
// Driver code
 
        let a = 5, b = 3, c = 2;
           
        // Function calling
        parabola(a, b, c);
             
            // This code is contributed by code_hunt.
</script>

Output –

Vertex:(-0.3, 1.55)
Focus: (-0.3, 1.6)
Directrix: y=-198

Time Complexity: O(1)

Auxiliary Space: O(1)

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