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. None of these

Answer 1

The correct option is C

$0$

Explanation for the correct option:

Find 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 $x-y=y$ in the above function, we get

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

Now, interchange the above function we get,

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

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

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

Hence, the correct option is C.