Question

# Consider the following function $f(x+2y,x-2y)=xy$, then $f(x,y)$ equals\begin{align} & \left( A \right)\dfrac{{{x}^{2}}-{{y}^{2}}}{8} \\ & \left( B \right)\dfrac{{{x}^{2}}-{{y}^{2}}}{4} \\ & \left( C \right)\dfrac{{{x}^{2}}+{{y}^{2}}}{4} \\ & \left( D \right)\dfrac{{{x}^{2}}-{{y}^{2}}}{2} \\ \end{align}

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.