What you can say with regular hiperbolas?Is equilatera hiperbolas or hiperbolas where the assymtotes are perpendiculars?

Relabel $\alpha, \beta, \gamma \longrightarrow a, b, c.$ $f(x,y) = y^2 - x^2 + 2pxy + qx + ry + s = 0$ is equation of general rectangular hyperbola $\mathcal R.$ For $y=0,$ $f(x,0) = -x^2 + qx + s = 0$ has roots $a, b$ $\Longrightarrow$ $q = a + b$ and $s = -ab.$ Substituting $(x,y) = (0,c)$ then yields $r = \frac{ab}{c} - c$ $\Longrightarrow$ $\mathcal R:\ y^2 - x^2 + 2pxy + (a+b)x + \left(\frac{ab}{c} - c\right)y - ab = 0,$ where $p$ is arbitrary. Let $H = (0,h)$ be orthocenter of $\triangle ABC.$ By power of the orthocenter $H$ WRT circumcircle of $\triangle ABC,$ $ab = -hc$ $\Longrightarrow$ $H = \left(0, -\frac{ab}{c}\right).$ For $x = 0,$ $f(0,y) = y^2 + ry + s = y^2 + \left(\frac{ab}{c} - c\right)y - ab = 0$ has 2 roots. One root is $y = c$ $\Longrightarrow$ the other root is $ y = -\frac{ab}{c}$ $\Longrightarrow$ $H \in \mathcal R.$ Rectangular hyperbola $\mathcal R$ cuts circumcircle of $\triangle ABC$ at $A, B, C$ and at one other point $D$ $\Longrightarrow$ $\mathcal R$ also goes through orthocenter $K$ of $\triangle DBC.$ $ADKH$ is a parallelogram with $AH \parallel DK$ inscribed in $\mathcal R$ $\Longrightarrow$ its diagonal intersection $P$ is the hyperbola center. Due to central similarity of the 9-point circle and circumcircle of $\triangle ABC$ with center $H$ and coefficient $\frac{_1}{^2},$ midpoint $P$ of $HD$ is on the 9-point circle.