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 <bits/stdc++.h>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 notclass 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 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 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 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 pointfunction 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 pointfunction 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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