Approximate root of $\alpha x - \beta y$ over $\mathbb Z$ except origin

Question

Consider the polynomial $ f(x,y) = \alpha x^2 - \beta y^2 $

Prove or disprove: For any choice of $\alpha, \beta \in \mathbb R_0^+$, the polynomial $f$ gets arbitrarily close to $0$ over $\mathbb Z \setminus (0,0)$

This obviously true for $\alpha$ and $\beta$ being square numbers.

Answer 1

Hint: for $x,y \in \Bbb Z$ and $\epsilon > 0$, we have $$ |f(x,y)| < \epsilon \iff\\ \left|\alpha x^2 - \beta y^2\right| < \epsilon \iff\\ \left|\frac{\alpha}{\beta} - \frac{y^2}{x^2}\right| < \frac{\epsilon}{\beta x^2} $$