Curve Tracing: Why I'm getting asymptotes parallel to both X & Y Axis?

Question

Here is equation of curve. $ x^2 y^2=a ^2 (y ^2- x^2) $

Here is my calculations...tell me if something is wrong in it. My Calculations

Answer 1

Solving for $x^2$ and $y^2$ we have

$$ x^2 = \frac{a^2 y^2}{a^2+ y^2}\\ y^2 = \frac{a^2 x^2}{a^2-x^2} $$

In the first, we have

$$ \lim_{y\to\infty} x^2 = a^2 $$

and in the second

$$ \lim_{x_{\pm}\to \pm a} = \pm\infty $$

so the asymptotes are located at $x = \pm a$