Question

If $f(x+2y,x-2y)=xy$, then $f(x,y)$ is equal to:

• A. $\frac{x^2-y^2}{8}$
• B. $\frac{x^2-y^2}{4}$
• C. $\frac{x^2+y^2}{4}$
• D. $\frac{x^2-y^2}{2}$

Answer 1

The correct option is A

$\frac{x^2-y^2}{8}$

Explanation for the correct answer.

Step 1: Find $x$and $y$ in terms of new variables

It is given that $f(x+2y,x-2y)=xy$

Let, $x+2y=p$ and $x-2y=q$

So,

$\begin{array}{rcl}p+q& =& x+2y+x-2y\\ \Rightarrow x& =& \frac{p+q}{2}\end{array}$

and

$\begin{array}{rcl}p-q& =& x+2y-x+2y\\ \Rightarrow y& =& \frac{p-q}{4}\end{array}$

Step 2: Find the value of $f(x,y)$

$\begin{array}{rcl}f(p,q)& =& \frac{p+q}{2}\times \frac{p-q}{4}\\ & =& \frac{p^2-q^2}{8}\end{array}$

Therefore, $\begin{array}{rcl}f(x,y)& =& \frac{x^2-y^2}{8}\end{array}$.

Hence, option A is correct.