Find Tangent at a given point on the curve

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Given a curve [ y = x(A – x) ], the task is to find tangent at given point (x, y) on that curve, where A, x, y are integers.
Examples:

Input: A = 2, x = 2, y = 0
Output: y = -2x - 4
Since y = x(2 - x)
      y = 2x - x^2 differentiate it with respect to x
      dy/dx = 2 - 2x  put x = 2, y = 0 in this equation
      dy/dx = 2 - 2* 2 = -2
      equation  => (Y - 0 ) = ((-2))*( Y - 2)
                => y = -2x -4

Input: A = 3, x = 4, y = 5
Output: Not possible
    Point is not on that curve

Approach:

First find if the given point is on that curve or not.
If the point is on that curve then, Find the derivative
Calculate the gradient of the tangent by Putting x, y in dy/dx.
Determine the equation of the tangent by substituting the gradient of the tangent and the coordinates of the given point into the gradient-point form of the straight line equation, where Equation of normal is Y – y = ( dy/dx ) * (X – x).

Below is the implementation of the above approach:

C++

// C++ program for find Tangent
// on a curve at given point
   
#include <bits/stdc++.h>
using namespace std;
   
// function for find Tangent
void findTangent(int A, int x, int y)
{
    // differentiate given equation
    int dif = A - x * 2;
   
    // check that point on the curve or not
    if (y == (2 * x - x * x)) {
   
        // if differentiate is negative
        if (dif < 0)
            cout << "y = "
                 << dif << "x" << (x * dif) + (y);
   
        else if (dif > 0)
   
            // differentiate is positive
            cout << "y = "
                 << dif << "x+" << -x * dif + y;
   
        // differentiate is zero
        else
            cout << "Not possible";
    }
}
   
// Driver code
int main()
{
    // declare variable
    int A = 2, x = 2, y = 0;
   
    // call function findTangent
    findTangent(A, x, y);
   
    return 0;
}

Java

// Java program for find Tangent
// on a curve at given point
import java.util.*;
import java.lang.*;
import java.io.*;
   
class GFG
{
    
// function for find Tangent
static void findTangent(int A, int x, int y)
{
    // differentiate given equation
    int dif = A - x * 2;
    
    // check that point on the curve or not
    if (y == (2 * x - x * x)) {
    
        // if differentiate is negative
        if (dif < 0)
            System.out.println( "y = "
                 + dif + "x" + (x * dif + y));
    
        else if (dif > 0)
    
            // differentiate is positive
            System.out.println( "y = "
                 + dif + "x+" + -x * dif + y);
    
        // differentiate is zero
        else
            System.out.println("Not possible");
    }
}
    
// Driver code
public static void main(String args[])
{
    // declare variable
    int A = 2, x = 2, y = 0;
    
    // call function findTangent
    findTangent(A, x, y);
    
}  
}

Python3

# Python3 program for find Tangent 
# on a curve at given point 
   
# function for find Tangent
def findTangent(A, x, y) :
   
    #  differentiate given equation 
    dif = A - x * 2
   
    #  check that point on the curve or not 
    if y == (2 * x - x * x) :
   
        # if differentiate is negative 
        if dif < 0 :
   
            print("y =",dif,"x",(x * dif) + (y))
   
        # differentiate is positive 
        elif dif > 0 :
   
            print("y =",dif,"x+",-x * dif + y)
   
        # differentiate is zero 
        else :
               
            print("Not Possible")
               
   # Driver code     
if __name__ == "__main__" :
   
    # declare variable 
    A, x, y = 2, 2, 0
   
    # call function findTangent
    findTangent(A, x, y)
                    
# This code is contributed by 
# ANKITRAI1

C#

// C# program for find Tangent 
// on a curve at given point 
    
using System; 
class GFG 
{ 
      
// function for find Tangent 
static void findTangent(int A, int x, int y) 
{ 
    // differentiate given equation 
    int dif = A - x * 2; 
      
    // check that point on the curve or not 
    if (y == (2 * x - x * x)) { 
      
        // if differentiate is negative 
        if (dif < 0) 
            Console.Write( "y = "
                 + dif + "x" + (x * dif + y)+"\n"); 
      
        else if (dif > 0) 
      
            // differentiate is positive 
            Console.Write( "y = "
                 + dif + "x+" + -x * dif + y+"\n"); 
      
        // differentiate is zero 
        else
            Console.Write("Not possible"+"\n"); 
    } 
} 
      
// Driver code 
public static void Main() 
{ 
    // declare variable 
    int A = 2, x = 2, y = 0; 
      
    // call function findTangent 
    findTangent(A, x, y); 
      
}   
}

PHP

<?php
// PHP program for find Tangent 
// on a curve at given point 
   
// function for find Tangent 
function findTangent($A, $x, $y) 
{ 
    // differentiate given equation 
    $dif = $A - $x * 2; 
   
    // check that point on the 
    // curve or not 
    if ($y == (2 * $x - $x * $x))
    { 
   
        // if differentiate is negative 
        if ($dif < 0) 
            echo "y = ", $dif , "x" , 
                  ($x * $dif) + ($y); 
   
        else if ($dif > 0) 
   
            // differentiate is positive 
            echo "y = ",
                $dif , "x+" , -$x * $dif + $y; 
   
        // differentiate is zero 
        else
            echo "Not possible"; 
    } 
} 
   
// Driver code 
   
// declare variable 
$A = 2;
$x = 2;
$y = 0; 
   
// call function findTangent 
findTangent($A, $x, $y); 
   
// This code is contributed by Sachin 
?>

Javascript

<script>
 
// javascript program for find Tangent
// on a curve at given point
    
     
// function for find Tangent
function findTangent( A,  x,  y) 
{ 
    // differentiate given equation
    var dif = A - x * 2; 
      
    // check that point on the curve or not
    if (y == (2 * x - x * x)) { 
      
        // if differentiate is negative
        if (dif < 0) 
            document.write( "y = "
                 + dif + "x" + (x * dif + y)+"\n"); 
      
        else if (dif > 0) 
      
            // differentiate is positive
            document.write( "y = "
                 + dif + "x+" + -x * dif + y+"\n"); 
      
        // differentiate is zero
        else
            document.write("Not possible"+"\n"); 
    } 
} 
      
// Driver code
 
    // declare variable
    var A = 2, x = 2, y = 0; 
      
    // call function findTangent
    findTangent(A, x, y); 
 
// This code is contributed by bunnyram19.
</script>

Output:

y = -2x-4

Time Complexity : O(1) ,as we are not using any loop.

Auxiliary Space : O(1) ,as we are not using any extra space.

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