Question

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

• A. $\frac{1}{4}xy$
• B. $\frac{1}{4}\left(x^{2}-y^{2}\right)$
• C. $\frac{1}{8}\left(x^{2}-y^{2}\right)$
• D. $\frac{1}{2}\left(x^{2}+y^{2}\right)$

Answer 1

Answer: (C)

Given $f(x+2 y, x-2 y)=x y$...(i)
Let $a=x+2 y$ and $b=x-2 y$
$\therefore a+b=2 x$ and $a-b=4 y$
From Eq. (i), we get
$ f(a, b) =\left(\frac{a+b}{2}\right)\left(\frac{a-b}{4}\right) $
$=\frac{a^{2}-b^{2}}{8} $
$ \therefore f(x, y) =\frac{x^{2}-y^{2}}{8} $