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