Reflection in a hyperbolic line formula

Question

Let $H$ denote the upper half-plane model of hyperbolic space. If $L$ is the hyperbolic line in $H$ given by a Euclidean semicircle with centre $a\in \mathbb{R}$ and radius $r >0$, show that reflection in the line $L$ is given by the formula $R_{L}(z)=a+ {r^2 \over{\overline{z} -a}} $.

Answer 1

Hint: $\operatorname{Isom}(\mathbb{H}) \cong SL_2(\mathbb{R})$ acting on $\mathbb{H}$ by Möbius transformations. Identify the Möbius transformation that preserves $L$ and swaps $a$ and $\infty$.

Answer 2

Hint: show that the circle inversion at the origin is $I(z)=\frac{r^2}{\overline{z}}$. To produce what you describe, you could first translate everything to the origin, reflect over the circle, and then translate back.