Answer

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**Hint:**For solving this question, we consider the given variables x+2y and x-2y as some other variables X and Y. Then we get two equations in terms of x, y and X, Y. Using those equations, we can find the values of x and y in terms of X and Y. Then we will get the function with variables X and Y only in terms of X and Y, by substituting them in place of x and y. Then we change X as x and Y as y to modify them to get an answer as in the options.

**Complete step by step answer:**

We were given a function f such that $f(x+2y,x-2y)=xy$.

Now let us consider two new variables X, Y such that

X=x+2y

Y=x-2y

Now, let us find the value of x in terms of X and Y.

Let us consider the value of X+Y.

$\begin{align}

& \Rightarrow X+Y=\left( x+2y \right)+\left( x-2y \right) \\

& \Rightarrow X+Y=2x \\

\end{align}$

So, we can write x in terms of X and Y as

$\begin{align}

& \Rightarrow X+Y=2x \\

& \Rightarrow x=\dfrac{X+Y}{2} \\

\end{align}$

Now, let us find the value of y in terms of X and Y.

Let us consider the value of X-Y.

$\begin{align}

& \Rightarrow X-Y=\left( x+2y \right)-\left( x-2y \right) \\

& \Rightarrow X-Y=4y \\

\end{align}$

So, we can write y in terms of X and Y as

$\begin{align}

& \Rightarrow X-Y=4y \\

& \Rightarrow y=\dfrac{X-Y}{4} \\

\end{align}$

So, the values of x and y in terms of X and Y are as below

$x=\dfrac{X+Y}{2}$

$y=\dfrac{X-Y}{4}$

So, we substitute these values in the given function value $f(x+2y,x-2y)=xy$.

Then, we get the function in the terms of X and Y.

$\begin{align}

& \Rightarrow f(x+2y,x-2y)=xy \\

& \Rightarrow f(X,Y)=\left( \dfrac{X+Y}{2} \right)\left( \dfrac{X-Y}{4} \right) \\

& \Rightarrow f(X,Y)=\dfrac{\left( X+Y \right)\left( X-Y \right)}{8} \\

\end{align}$

Now, let us consider the formula,

$\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$

So, the value of product of (X+Y) and (X-Y) is

$\left( X+Y \right)\left( X-Y \right)={{X}^{2}}-{{Y}^{2}}$

Then, we can write the function as,

\[\Rightarrow f(X,Y)=\dfrac{{{X}^{2}}-{{Y}^{2}}}{8}\]

So, for the function f on X and Y value of the function is \[f(X,Y)=\dfrac{{{X}^{2}}-{{Y}^{2}}}{8}\].

Now, let us replace X by x and Y by y, then we can write the function as,

\[\Rightarrow f(x,y)=\dfrac{{{x}^{2}}-{{y}^{2}}}{8}\]

**So, the correct answer is “Option A”.**

**Note:**The chance of occurrence of mistake is at the ending of the solution, one might think that we should not change the variables X and Y into x and y. Here, we are not transforming the variables like we did in the starting of the solution, we are just changing the symbol from X to x and Y to y to make it look like the one in the given options.

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