Question

If $f(x+y,x-y)=xy$, then $\frac{f(x,y)+f(y,x)}{2}$ is

• A. $x$
• B. $y$
• C. $0$
• D. $\frac{x+y}{2}$

Answer 1

The correct option is C $0$

Assuming $x+y=u,x-y=v$
$\Rightarrow x=\frac{u+v}{2},y=\frac{u-v}{2}$

$f(x+y,x-y)=xy$
$\Rightarrow f(u,v)=\left(\frac{u+v}{2}\right)\times \left(\frac{u-v}{2}\right)$
$\Rightarrow f(u,v)=\frac{u^2-v^2}{4}$
$\Rightarrow f(u,v)+f(v,u)=\frac{u^2-v^2}{4}+\frac{v^2-u^2}{4}\Rightarrow f(u,v)+f(v,u)=0$
$\Rightarrow f(x,y)+f(y,x)=0$

$\therefore \frac{f(x,y)+f(y,x)}{2}=0$