Maximum points of intersections possible among X circles and Y straight lines

Given two integers X and Y, the task is to find the maximum number of points of intersection possible among X circles and Y straight lines.

Example:

Input: X = 4, Y = 4
Output: 50
Explanation:
4 lines intersect each other at 6 points and 4 circles intersect each other at maximum of 12 points.
Each line intersects 4 circles at 8 points.
Hence, 4 lines intersect four circles at a maximum of 32 points.
Thus, required number of intersections = 6 + 12 + 32 = 50.

Input: X = 3, Y = 4
Output: 36

Approach:
It can be observed that there are three types of intersections:



  1. The number of ways to choose a pair of points from X circles is \binom{X}{2}. Each such pair intersect at most two points.
  2. The number of ways to choose a pair of points from Y lines is \binom{Y}{2}. Each such pair intersect in at most one point.
  3. The number of ways to choose one circle and one line from X circles and Y lines is is X*Y. Each such pair intersect in at most two points.

So, the maximum number of point of intersection can be calculated as:
=> 2*\binom{X}{2} + \binom{Y}{2} + 2*X*Y
=> X*(X - 1) + \frac{Y*(Y - 1)}{2} + 2*X*Y

Thus, formula to find maximum number of point of intersection of X circles and Y straight lines is:
\frac{Y*(Y - 1)}{2} + X*(2*Y + X - 1)

Below is the implementation of the above approach:

C++

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// C++ Program of the above approach
#include <bits/stdc++.h>
using namespace std;
  
int maxPointOfIntersection(int x, int y) 
{
    int k = y * (y - 1) / 2;
    k = k + x * (2 * y + x - 1);
    return k;
}
  
// Driver code
int main()
{
      
    // Number of circles
    int x = 3;
      
    // Number of straight lines
    int y = 4;
  
    // Function Call
    cout << (maxPointOfIntersection(x, y));
}
  
// This code is contributed by Ritik Bansal

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Java

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// Java program to implement
// the above approach
class GFG{
      
static int maxPointOfIntersection(int x, int y) 
{
    int k = y * (y - 1) / 2;
    k = k + x * (2 * y + x - 1);
    return k;
}
  
// Driver code
public static void main(String[] args)
{
      
    // Number of circles
    int x = 3;
      
    // Number of straight lines
    int y = 4;
  
    // Function Call
    System.out.print(maxPointOfIntersection(x, y));
}
}
  
// This code is contributed by Princi Singh

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Python3

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# Python3 program to implement
# the above approach
def maxPointOfIntersection(x, y):
    k = y * ( y - 1 ) // 2
    k = k + x * ( 2 * y + x - 1 )
    return
  
# Number of circles
x = 3
# Number of straight lines
y = 4
  
# Function Call
print(maxPointOfIntersection(x, y))

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C#

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// C# program to implement
// the above approach
using System;
  
class GFG{
      
static int maxPointOfIntersection(int x, int y) 
{
    int k = y * (y - 1) / 2;
    k = k + x * (2 * y + x - 1);
    return k;
}
  
// Driver code
public static void Main(String[] args)
{
      
    // Number of circles
    int x = 3;
      
    // Number of straight lines
    int y = 4;
  
    // Function Call
    Console.Write(maxPointOfIntersection(x, y));
}
}
  
// This code is contributed by Princi Singh

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Output:

36

Time Complexity: O(1)
Auxiliary Space: O(1)

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Improved By : princi singh, bansal_rtk_