 Solution

A Geodesic in the Poincare half-plane model is a vertical line or a semicircle orthogonal to the real axis. Given a hyperbolic line $\ell$, choose a real Möbius transformation sending its two ideal endpoints to $0$ and $\infty$. After multiplying its representing matrix by a scalar and, if necessary, changing its sign, it represents an element of $PS{L}_2(\mathbb{R})$ and sends $\ell$ to the positive imaginary axis ${\ell }_{+}$.

The chosen point $P\in \ell$ then maps to $iy$ for some $y>0$. The dilation $z\mapsto z/y$, represented byfixes ${\ell }_{+}$ and sends $iy$ to $i$. Thus every pointed line $(\ell ,P)$ can be sent to $({\ell }_{+},i)$, proving transitivity on pointed hyperbolic lines.