Q41 / 87 · NTA Abhyas (JEE Main) 2020 · Mathematics · Functions

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

A
$x$
B
$y$
Correct answer
C
$0$
D
$\frac{x^{2} - y^{2}}{2}$

Solution

Let,

x + y = p, x - y = q

Then,

f (p, q) = \frac{p + q}{2} \cdot \frac{p - q}{2} = \frac{p^{2} - q^{2}}{4}

∴ f (x, y) = \frac{x^{2} - y^{2}}{4}

and

f (y, x) = \frac{y^{2} - x^{2}}{4}

A . M . = \frac{f (x, y) + f (y, x)}{2}

= \frac{\frac{x^{2} - y^{2}}{4} + \frac{y^{2} - x^{2}}{4}}{2} = 0

Hence, the correct answer is option: 'C'.

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