This question it is for students of an high school. Supposing that I have a homographic function, with the condition $\Delta=bc-ad\neq 0$ with $a,b,c,d\in\mathbb{R}$ with $cx+d\neq 0$,
$$y=\frac{ax+b}{cx+d} \tag 1$$
We know that the asymptotes are the straight lines of equations:
$$a_1\colon\, x=-\frac dc, \quad a_2\colon\, y=\frac ac$$ obviously using the translation vector $\mathbf{v}=(-d/c;a/c)$, where the $(1)$ becomes
$$XY=\frac{\Delta}{c^2}:=\lambda, \quad \lambda\in \Bbb R$$
If the vertexes for the rotated equilateral hyperbola equation $x^2-y^2=a^2 \to xy=k$ are $V\equiv (\pm \sqrt{|k|}, \pm \sqrt{|k|})$ and the foci are $F\equiv (\pm \sqrt{2|k|}, \pm \sqrt{2|k|})$, if $k>0$ and $V\equiv (\mp\sqrt{|k|}, \pm \sqrt{|k|})$ and the foci are $F\equiv (\mp \sqrt{2|k|}, \pm \sqrt{2|k|})$ if $k<0$, are the vertexes of the $(1)$ (i.e. foci)
$$V\equiv \left(-\frac dc\pm \sqrt{|k|}, \frac ac\pm \sqrt{|k|}\right), \quad F\equiv \left(-\frac dc\pm \sqrt{2|k|}, \frac ac\pm \sqrt{2|k|}\right), \quad k>0$$ and
$$V\equiv \left(-\frac dc\mp \sqrt{|k|}, \frac ac\pm \sqrt{|k|}\right), \quad F\equiv \left(-\frac dc\mp \sqrt{2|k|}, \frac ac\pm \sqrt{2|k|}\right), \quad k<0, \quad \color{red}{?}$$
Are there other methods to find the foci or is this standard procedure used? I hope I haven't written it incorrectly and I look for your welcome answers.