Prove that if a, b, c are integers and x, y, z are non-integer real numbers and $\alpha$ is a real number, for every given set of x, y, z, the number $\alpha$ obtained from the following equation:
$$\frac{a^2}{x^2} + \frac{b^2}{y^2} + \frac{c^2}{z^2} = \alpha$$
cannot be obtained from other combination of a, b and c. In other words, every natural number $\alpha$ that satisfies the above equation can only be found from a unique set of a, b and c.