# Find normal at a given point on the curve

Given a curve [ y = x(A – x) ], the task is to find normal at a given point ( x, y) on that curve, where A is an integer number and x, y also any integer.

Examples:

```Input: A = 2, x = 2, y = 0
Output: 2y = x - 2
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 ) = ((-1/-2))*( Y - 2)
=> 2y = x -2

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

Approach: First we need to find given point is on that curve or not if the point is on that curve then:

1. We need to differentiate that equation that point don’t think too much for differentiation of this equation if you analyze then you find that dy/dx always become A – 2x.
2. Put x, y in dy/dx.
3. Equation of normal is Y – y = -(1/( dy/dx )) * (X – x).

Below is the implementation of the above approach:

## C++

 `// C++ program for find curve` `// at given point` `#include ` `using` `namespace` `std;`   `// function for find normal` `void` `findNormal(``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 << 0 - dif << ``"y = "` `                 ``<< ``"x"` `<< (0 - x) + (y * dif);`   `        ``else` `if` `(dif > 0)`   `            ``// differentiate is positive` `            ``cout << dif << ``"y = "` `                 ``<< ``"-x+"` `<< x + dif * y;`   `        ``// differentiate  is zero` `        ``else` `            ``cout << ``"x = "` `<< x;` `    ``}`   `    ``// other wise normal not found` `    ``else` `        ``cout << ``"Not possible"``;` `}`   `// Driver code` `int` `main()` `{` `    ``// declare variable` `    ``int` `A = 2, x = 2, y = 0;`   `    ``// call function findNormal` `    ``findNormal(A, x, y);` `    ``return` `0;` `}`

## Java

 `// Java program for find curve` `// at given point`   `import` `java.io.*;`   `class` `GFG {`   `// function for find normal` `static` `void` `findNormal(``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.print( (``0` `- dif) + ``"y = "` `                ``+ ``"x"` `+((``0` `- x) + (y * dif)));`   `        ``else` `if` `(dif > ``0``)`   `            ``// differentiate is positive` `            ``System.out.print( dif + ``"y = "` `                ``+ ``"-x+"` `+ (x + dif * y));`   `        ``// differentiate is zero` `        ``else` `            ``System.out.print( ``"x = "` `+x);` `    ``}`   `    ``// other wise normal not found` `    ``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 findNormal` `    ``findNormal(A, x, y);;` `    ``}` `}` `// This Code is contributed by inder_verma..`

## Python3

 `# Python 3 program for find curve` `# at given point`   `# function for find normal` `def` `findNormal(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``(``0` `-` `dif, ``"y ="``, ``"x"``, ` `                 ``(``0` `-` `x) ``+` `(y ``*` `dif))`   `        ``elif` `(dif > ``0``):` `            `  `            ``# differentiate is positive` `            ``print``(dif, ``"y ="``, ``"- x +"``, ` `                        ``x ``+` `dif ``*` `y)`   `        ``# differentiate is zero` `        ``else``:` `            ``print``(``"x ="``, x)`   `    ``# other wise normal not found` `    ``else``:` `        ``print``(``"Not possible"``)`   `# Driver code` `if` `__name__ ``=``=` `'__main__'``:` `    `  `    ``# declare variable` `    ``A ``=` `2` `    ``x ``=` `2` `    ``y ``=` `0`   `    ``# call function findNormal` `    ``findNormal(A, x, y)` `    `  `# This code is contributed By` `# Surendra_Gangwar`

## C#

 `// C# program for find curve ` `// at given point ` `using` `System;`   `class` `GFG` `{` `    `  `// function for find normal` `static` `void` `findNormal(``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((0 - dif) + ``"y = "` `+ ` `                   ``"x"` `+ ((0 - x) + (y * dif)));`   `        ``else` `if` `(dif > 0)`   `            ``// differentiate is positive` `            ``Console.Write(dif + ``"y = "` `+ ` `                          ``"-x + "` `+ (x + dif * y));`   `        ``// differentiate is zero` `        ``else` `            ``Console.Write(``"x = "` `+ x);` `    ``}`   `    ``// other wise normal not found` `    ``else` `        ``Console.WriteLine(``"Not possible"``);` `}`   `// Driver code` `static` `public` `void` `Main ()` `{` `    ``// declare variable` `    ``int` `A = 2, x = 2, y = 0;` `    `  `    ``// call function findNormal` `    ``findNormal(A, x, y);` `}` `}`   `// This code is contributed by ajit`

## PHP

 ` 0) `   `            ``// differentiate is positive ` `            ``echo` `\$dif` `, ``"y = "``,` `                 ``"-x+"` `,( ``\$x` `+ ``\$dif` `* ``\$y``); `   `        ``// differentiate is zero ` `        ``else` `            ``echo` `"x = "` `, ``\$x``; ` `    ``} `   `    ``// other wise normal not found ` `    ``else` `        ``echo` `"Not possible"``; ` `} `   `// Driver code `   `// declare variable ` `\$A` `= 2;` `\$x` `= 2;` `\$y` `= 0; `   `// call function findNormal ` `findNormal(``\$A``, ``\$x``, ``\$y``); `   `// This code is contributed by ajit` `?>`

## Javascript

 ``

Output:

`2y = x-2`

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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