explicit formula for hyperbolic translation

Question

Let $S(a,r)$ be a sphere around $a \in \mathbb{R}^n$ of radius $r \in \mathbb{R}_+$. Assume further that this sphere is orthogonal to $S^{n-1}$, i.e. $r^2 = |a|^2-1$. Let $\sigma_a$ be the reflection in $S(a,r)$, namely:
\[\sigma_a(x) = a + \frac{r^2 \cdot (x-a)}{|x-a|^2}.\] Let $\rho_a$ be the reflection in the plane $\{x \in \mathbb{R}^n | a \cdot x = 0 \}$, namely:
\[\rho_a(x) = x -2(a\cdot x)a\] where by $\cdot$ between two vectors I mean the standard scalar product. Show that:
\[\sigma_a \circ \rho_a(x) = \frac{(|a|^2-1)x + (|x|^2+2x\cdot a^*+1)a}{|x+a|^2}\] where $a^* := \frac{a}{|a|^2}$.
This is left to the reader on page 124 of Foundations of Hyperbolic Manifolds. By John G.Ratcliffe. This then leads to the formula for hyperbolic translation. But I can't seem to even get the right denominator.