Check if a point is inside, outside or on the parabola

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Given a parabola with vertex (h, k), and $a$ , the distance between focus and vertex. The task is to determine if the point (x, y) is inside, outside or on the parabola.
Examples:

Input: h = 100, k = 500, x = 20, y = 10, a = 4 
Output: Outside

Input: h = 0, k = 0, x = 2, y = 1, a = 4
Output: Inside

Approach: It is very simple, we have to just solve the equation for point (x, y):

(y-k)^2 = 4a(x-h)
or, (y-k)^2 – 4a(x-h) = 0

After solving, if the result comes less than 0 then the point lies within, else if it comes exact 0 then the point lies on the parabola, and if the result is greater than 0 unsatisfied the point lies outside of the parabola.
Here we are taking a parabola whose axis of symmetry is y = k, although the approach is applicable for any parabola.
Below is the implementation of above approach:

C++

// C++ Program to check if the point
// lies within the parabola or not
#include <bits/stdc++.h>
using namespace std;
 
// Function to check the point
int checkpoint(int h, int k, int x, int y, int a)
{
 
    // checking the equation of
    // parabola with the given point
    int p = pow((y - k), 2) - 4 * a * (x - h);
 
    return p;
}
 
// Driver code
int main()
{
    int h = 0, k = 0, x = 2, y = 1, a = 4;
 
    if (checkpoint(h, k, x, y, a) > 0)
        cout << "Outside" << endl;
 
    else if (checkpoint(h, k, x, y, a) == 0)
        cout << "On the parabola" << endl;
 
    else
        cout << "Inside" << endl;
 
    return 0;
}

Java

// Java Program to check if the point
// lies within the parabola or not
 
class solution
{
 
// Function to check the point
static int checkpoint(int h, int k, int x, int y, int a)
{
    // checking the equation of
    // parabola with the given point
    int p =(int) Math.pow((y - k), 2) - 4 * a * (x - h);
 
    return p;
}
 
//driver code 
public static void main(String arr[])
{
 
    int h = 0, k = 0, x = 2, y = 1, a = 4;
 
    if (checkpoint(h, k, x, y, a) > 0)
    System.out.println("Outside");
 
    else if (checkpoint(h, k, x, y, a) == 0)
    System.out.println("On the parabola");
 
    else
    System.out.println("Inside");
 
}
}

Python3

# Python3 Program to check if the point
# lies within the parabola or not
 
#  Function to check the point 
def checkpoint(h, k, x, y, a):
 
    # checking the equation of
    # parabola with the given point
    p = pow((y - k), 2) - 4 * a * (x - h)
 
    return p
 
# Driver code
if __name__ == "__main__" : 
     
    h = 0
    k = 0
    x = 2
    y = 1
    a = 4
 
    if checkpoint(h, k, x, y, a) > 0:
        print ("Outside\n")
 
    elif checkpoint(h, k, x, y, a) == 0:
        print ("On the parabola\n")
 
    else:
        print ("Inside\n");
         
# This code is contributed by 
# Surendra_Gangwar

C#

// C# Program to check if the point 
// lies within the parabola or not 
using System;
 
class GFG
{
 
// Function to check the point 
public static int checkpoint(int h, int k,
                             int x, int y, 
                             int a)
{
    // checking the equation of 
    // parabola with the given point 
    int p = (int) Math.Pow((y - k), 2) - 
                            4 * a * (x - h);
 
    return p;
}
 
// Driver code 
public static void Main(string[] arr)
{
    int h = 0, k = 0, 
        x = 2, y = 1, a = 4;
 
    if (checkpoint(h, k, x, y, a) > 0)
    {
        Console.WriteLine("Outside");
    }
 
    else if (checkpoint(h, k, x, y, a) == 0)
    {
        Console.WriteLine("On the parabola");
    }
 
    else
    {
        Console.WriteLine("Inside");
    }
}
}
 
// This code is contributed 
// by Shrikant13

PHP

<?php
// PHP Program to check if 
// the point lies within 
// the parabola or not
 
// Function to check the point
function checkpoint($h, $k, $x, 
                        $y, $a)
{
 
    // checking the equation of
    // parabola with the given point
    $p = pow(($y - $k), 2) - 4 * 
              $a * ($x - $h);
 
    return $p;
}
 
// Driver code
$h = 0; $k = 0; $x = 2;
$y = 1; $a = 4;
 
if (checkpoint($h, $k, $x, 
               $y, $a) > 0)
    echo "Outside";
else if (checkpoint($h, $k, $x, 
                    $y, $a) == 0)
    echo "On the parabola";
else
    echo "Inside";
 
// This code is contributed 
// by inder_verma
?>

Javascript

<script>
 
// javascript Program to check if the point
// lies within the parabola or not
 
// Function to check the point
function checkpoint(h , k , x , y , a)
{
    // checking the equation of
    // parabola with the given point
    var p =parseInt(Math.pow((y - k), 2) - 4 * a * (x - h));
 
    return p;
}
 
//driver code 
var h = 0, k = 0, x = 2, y = 1, a = 4;
 
if (checkpoint(h, k, x, y, a) > 0)
    document.write("Outside");
 
else if (checkpoint(h, k, x, y, a) == 0)
    document.write("On the parabola");
 
else
    document.write("Inside");
 
// This code is contributed by 29AjayKumar 
 
</script>

Output:

Inside

Time Complexity: O(1)

Auxiliary Space: O(1)

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