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Finding the vertex, focus and directrix of a parabola
  • Difficulty Level : Basic
  • Last Updated : 22 May, 2018

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.

22

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 y=ax^2+bx+c
Vertex: (-b/2a, 4ac-b^2/4a)
Focus: (-b/2a, 4ac-b^2+1/4a)
Directrix: y=c-(b^2+1)4a

C++

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#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;
}

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Java

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// 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

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Python 3

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# 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.

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C#

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// 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

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PHP

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<?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.
?>

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Output –

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

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