 Hyperbolic lines meeting the imaginary axis at a fixed angle

The hyperbolic metric in the half-plane model is conformal, so hyperbolic and Euclidean angles agree. The line meeting the imaginary axis at $iy$, $y>0$, at signed angle $\alpha \in (0,\pi )$ is represented by the Euclidean circle with centre $y\cot \alpha$ on the real axis and radius $y\csc \alpha$.