Given a hyperbola centered at (h, k), with semi-major axis a, semi-minor axis b, both aligned with the Cartesian plane, the task is to determine if the point (x, y) lies within the area bounded by the hyperbola or not.
Examples:
Input: h = 0, k = 0, x = 2, y = 1, a = 4, b = 5
Output: InsideInput: h = 1, k = 2, x = 200, y = 100, a = 6, b = 5
Output: Outside
Approach: The given problem can be solved by solving the equation of hyperbola mentioned below, for the given point (x, y):
Based on the evaluated value of the above equation, the output of the program is as follows:
- Less than 1: The point lies inside the hyperbola.
- Equal to 1: The point lies on the hyperbola
- Greater than 1: The point lies outside the hyperbola.
Below is the implementation of the above approach:
C++
// C++ program for the// above approach#include <bits/stdc++.h>using namespace std;// Function to check if the point// (x, y) lies inside, on or// outside the given hyperbolavoid checkpoint(int h, int k, int x, int y, int a, int b){ // Stores the value of the equation int p = (pow((x - h), 2) / pow(a, 2)) - (pow((y - k), 2) / pow(b, 2)); // Generate output based on value of p if (p > 1) { cout << "Outside"; } else if (p == 1) { cout << "On the Hyperbola"; } else { cout << "Inside"; }}// Driver Codeint main(){ int h = 0, k = 0, x = 2; int y = 1, a = 4, b = 5; checkpoint(h, k, x, y, a, b); return 0;} |
Java
// Java program for the above approachimport java.io.*;import java.lang.*;import java.util.*;class GFG{// Function to check if the point// (x, y) lies inside, on or// outside the given hyperbolastatic void checkpoint(int h, int k, int x, int y, int a, int b){ // Stores the value of the equation int p = (int)(Math.pow((x - h), 2) / Math.pow(a, 2)) - (int)(Math.pow((y - k), 2) / Math.pow(b, 2)); // Generate output based on value of p if (p > 1) { System.out.println("Outside"); } else if (p == 1) { System.out.println("On the Hyperbola"); } else { System.out.println("Inside"); }}// Driver Codepublic static void main(String[] args){ int h = 0, k = 0, x = 2; int y = 1, a = 4, b = 5; checkpoint(h, k, x, y, a, b);}}// This code is contributed by Kingash |
Python3
# Python3 program for the above approachfrom math import pow# Function to check if the point# (x, y) lies inside, on or# outside the given hyperboladef checkpoint(h, k, x, y, a, b): # Stores the value of the equation p = ((pow((x - h), 2) // pow(a, 2)) - (pow((y - k), 2) // pow(b, 2))) # Generate output based on value of p if (p > 1): print("Outside") elif (p == 1): print("On the Hyperbola"); else: print("Inside")# Driver Codeif __name__ == "__main__": h = 0 k = 0 x = 2 y = 1 a = 4 b = 5 checkpoint(h, k, x, y, a, b)# This code is contributed by AnkThon |
C#
// C# program for the above approachusing System;class GFG{// Function to check if the point// (x, y) lies inside, on or// outside the given hyperbolastatic void checkpoint(int h, int k, int x, int y, int a, int b){ // Stores the value of the equation int p = (int)(Math.Pow((x - h), 2) / Math.Pow(a, 2)) - (int)(Math.Pow((y - k), 2) / Math.Pow(b, 2)); // Generate output based on value of p if (p > 1) { Console.WriteLine("Outside"); } else if (p == 1) { Console.WriteLine("On the Hyperbola"); } else { Console.WriteLine("Inside"); }}// Driver Codepublic static void Main(string[] args){ int h = 0, k = 0, x = 2; int y = 1, a = 4, b = 5; checkpoint(h, k, x, y, a, b);}}// This code is contributed by ukasp |
Javascript
<script>// JavaScript program for the above approach// Function to check if the point// (x, y) lies inside, on or// outside the given hyperbolafunction checkpoint(h, k, x, y, a, b){ // Stores the value of the equation p = ((Math.pow((x - h), 2) / Math.pow(a, 2)) - (Math.pow((y - k), 2) / Math.pow(b, 2))) // Generate output based on value of p if (p > 1) console.log("Outside"); else if (p == 1) console.log("On the Hyperbola"); else console.log("Inside");}// Driver Codefunction main(){ var h = 0; var k = 0; var x = 2; var y = 1; var a = 4; var b = 5; checkpoint(h, k, x, y, a, b); }// main function callmain()// This code is contributed by AnkThon</script> |
Output:
Inside
Time Complexity: O(1)
Auxiliary Space: O(1)
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