What is the value of $x*y$? 
Given that $$\left(\frac{x}{y}\right)^{-2} + \left(\frac{y}{x}\right)^{-2} = \frac{10}{3}$$
  find the value of $x*y$.

My question is, can we calculate the value of $x*y$ or not? If yes, then how? If not, then why?

I think you can using the following result:
$(a^2 + b^2) = (a+b)^2 - 2ab$
$$\left( \frac{x}{y}\right)^{-2} + \left(\frac{x}{y}\right)^{2} = \frac{10}{3} $$
$$\left(\frac{x}{y} + \frac{1}{\frac{x}{y}}\right)^2 - 2\times\frac{x}{y} \times \frac{1}{\frac{x}{y}} = \frac{10}{3}$$
After simplifying you obtain:
$\left(\frac{x^2 + y^2}{xy}\right)^2 = \frac{16}{3} $
Eventually, you get something thats looks like:
$$ \frac{x^2 + y^2}{xy} = \frac{4}{\sqrt3}$$
As pointed out in the other answers, there seems to be infinitely many solutions.
