The hyperbolic line $l=C\cap \mathbb{H}^2$ has center $(\frac{5}{4}, \frac{1}{4})$ and radius $r=\frac{\sqrt{10}}{4}$. Given two points $P=(\frac{5}{8}, -\frac{5}{8})$ and $Q=(\frac{3}{4},0)$, I need to find the hyperbolic line $m$ such that $\Omega_m\Omega_l(P) = Q$. Here $\Omega_l$ is the hyperbolic reflection with respect to $l$, which is just the inversion with respect to the Euclidean circle $C$ in $\mathbb{R}^2$.
My question is, are there easier ways to determine this hyperbolic line $m$? I tried this way, calculating the coordinate of $S=\Omega_l(P)$ using the formula $|O_CS|\cdot |O_CP|=r^2$. Then I need to find the center and radius of $m$, which involves so much more calculations to do. Am I missing something? Is there a simpler way to compute?