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 :

Solution :
$$\eqalign{ & {\text{Let }}x + y = p,\,\,x - y = q \cr & {\text{Then }}f\left( {p,\,q} \right) = \frac{{p + q}}{2}.\frac{{p - q}}{2} = \frac{{{p^2} - {q^2}}}{4} \cr & \therefore \,\,f\left( {x,\,y} \right) = \frac{{{x^2} - {y^2}}}{4}{\text{ and }}f\left( {y,\,x} \right) = \frac{{{y^2} - {x^2}}}{4} \cr} $$