If $$a = \frac{x}{{x + y}}$$   and $$b = \frac{y}{{x - y}}{\text{,}}$$   then $$\frac{{ab}}{{a + b}}$$   is equal to = ?

$$\eqalign{ & \frac{{ab}}{{a + b}} = \frac{{\frac{x}{{x + y}} \times \frac{y}{{x - y}}}}{{\frac{x}{{x + y}} + \frac{y}{{x - y}}}} \cr & \frac{{ab}}{{a + b}} = \frac{{\frac{{xy}}{{{x^2} - {y^2}}}}}{{\frac{{x\left( {x - y} \right) + y\left( {x + y} \right)}}{{{x^2} - {y^2}}}}} \cr & \frac{{ab}}{{a + b}} = \frac{{xy}}{{{x^2} + {y^2}}} \cr} $$