Question

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

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

Answer 1

Answer: (C)

$\frac{1}{8}(x^2-y^2)$

Given $f(x+2y,x-2y)=xy.......\left(1\right)$
Let $a=x+2y$ and $b=x-2y$
$\therefore$ $a+b=2x$ and $a-b=4y$
From Eq $\left(i\right)$ we get
$f(a,b)=\left(\frac{a+b}{2}\right)\left(\frac{a-b}{4}\right)$
$=\frac{a^2-b^2}{8}$
$\therefore$ $f(x,y)=\frac{x^2-y^2}{8}$