Hyperbolic lines meeting the imaginary axis at a fixed angle
= 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 $y\csc\alpha$.