<s>[INST] 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?
 [/INST] As @Meelo said, if (x,y) is a solution so is (cx,cy) when c is not equal to zero.
$$\frac {y^2}{x^2} + \frac {x^2}{y^2} = \frac {y^4+x^4}{x^2y^2}$$
$$\frac{(cy)^2}{(cx)^2}+\frac{(cx)^2}{(cy)^2} = \frac{c^4(y^4+x^4)}{c^4x^2y^2} = \frac{x^4+y^4}{x^2y^2}$$
$$cx\cdot cy = c^2xy$$
Thus we can not find any single value for $xy$ because there are infinite solutions.
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