Question

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

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

Answer 1

Answer: (C)

Let, $\text{x}+\text{y}=\text{p, x}-\text{y}=\text{q}$
Then, $\text{f}\left(\text{p, q}\right)=\frac{\text{p}+\text{q}}{2}\cdot \frac{\text{p}-\text{q}}{2}=\frac{{\left(\text{p}\right)}^2-{\left(\text{q}\right)}^2}{4}$
$\therefore \text{f}\left(\text{x, y}\right)=\frac{{\left(\text{x}\right)}^2-{\left(\text{y}\right)}^2}{4}$ and $\text{f}\left(\text{y, x}\right)=\frac{{\left(\text{y}\right)}^2-{\left(\text{x}\right)}^2}{4}$
$A.M.=\frac{f\left(x,y\right)+f\left(y,x\right)}{2}$
$=\frac{\frac{x^2-y^2}{4}+\frac{y^2-x^2}{4}}{2}=0$