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

• Last Updated : 23 Jul, 2022

Given a parabola with vertex (h, k), and , 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 using namespace std; // Function to check the pointint 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 codeint 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 pointstatic 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 codepublic 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 pointdef 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 codeif __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 notusing System; class GFG{ // Function to check the pointpublic 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 codepublic 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

 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



Output:

Inside

Time Complexity: O(1)

Auxiliary Space: O(1)

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