= 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 $PSL_2(\mathbb R)$ 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 by
$$
\begin{pmatrix}y^{-1/2}&0\\0&y^{1/2}\end{pmatrix}\in SL_2(\mathbb R),
$$
fixes $\ell_+$ and sends $iy$ to $i$. Thus every pointed line $(\ell,P)$ can be sent to $(\ell_+,i)$, proving <transitivity on pointed hyperbolic lines>.
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