Question

If $f\left(x+y,x-y\right)=xy$ ,then the arithmetic mean of $f\left(x,y\right)$ and $f\left(y,x\right)$ is

• A. $x$
• B. $y$
• C. $0$
• D. $1$

Answer 1

The correct option is C

$0$

Explanation for the correct option:

Finding the arithmetic mean of $f\left(x,y\right)$ and $f\left(y,x\right)$:

Given that,

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

Replace $x+y=x$ and $x-y=y$, we get

$\begin{array}{rcl}f\left(x,y\right)& =& \left(x+y\right)\left(x-y\right)\\ & =& x^2-y^2...\left(1\right)\end{array}$

Similarly, replace $x$ by $y$ in $\left(1\right)$,

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

Now, the arithmetic mean of $f\left(x,y\right)$ and $f\left(y,x\right)$ is

$\begin{array}{rcl}\frac{f\left(x,y\right)+f\left(y,x\right)}{2}& =& \frac{x^2-y^2+y^2-x^2}{2}\left[\mathrm{from}\left(1\right)\&\left(2\right)\right]\\ & =& 0\end{array}$

Hence, the correct option is C.