Question

If $f(x+2y,x-2y)=xy$, then $f(x,y)$ equals

• 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 $\frac{x^2-y^2}{8}$

$f\left(x+2y,x-2y\right)=xy$

Let $x+2y=U$ ----$\left(1\right)$

$x-2y=u$-----$\left(2\right)$

Adding $\left(1\right)$ & $\left(2\right)$

$\begin{array}{c}2x=U+u\\ x=\frac{U+u}{2}\end{array}$

Subtracting $\left(2\right)$ from $\left(1\right)$

$\begin{array}{c}4y=U-u\\ \Rightarrow y=\frac{U-u}{4}\\ f\left(U,u\right)=\left(\frac{U+u}{2}\right)\left(\frac{U-u}{4}\right)\\ =\frac{U^2-u^2}{8}\end{array}$

Hence

$f\left(x,y\right)=\frac{x^2-y^2}{8}$