In the following figure, $h(A,B)$ is an horocycle centered in A passing over B. $\Theta(h)$ is the angle of parallelism of the segment $h$ and $S$ is the well known intersection of a chord of an horocycle centered in $\Omega$ passing over the origin of the coordinate system with an angle of parallelism $\frac{\pi}{4}$.
I need to prove the following theorem:
In the horocycle $h(\Omega,O)$, consider a point P. Let $s$ be the measure of $PP_x$. Then
(i) If s = S then the line $ \Omega P $ is parallel to y;
(ii) If s > S then the line $ \Omega P $ is ultraparallel to y;
(iii) If s < S then the line $ \Omega P $ intersects y.
I have been thinking about your problem (which i think is more difficult than you should be required to solve).
I will solve it using the Poincare Half plane model ( https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model ) If your level is as high as it should be to solve this problem then you should also be able to transform it to the model of hyperbolic plane you are using.
The main points of the Poincare Half plane model are:
the model models the complete hyperbolic plane by using the upper half plane.
hyperbolic lines are represented by rays and halfcircles orthogonal to the x axis.
horocycles are represented by circles tangent to the x axis, their hyperbolic centre is the point where they touch the x axis.
hypercycles ( https://en.wikipedia.org/wiki/Hypercycle_%28hyperbolic_geometry%29 ) are are by lines and circle arcs that are not orthogonalto the x axis.
the model is conformal, the angles in the model have the same size as the angles on the hyperbolic plane.
The problem splits in a couple of sub proofs:
Proof that the line $\Omega S$ is limiting parallel to line $y$
Proof that if $P_xP < S_xS $ then the line $\Omega P$ will intersect line $y$
Proof that if $P_xP > S_xS $ then the line $\Omega P$ will intersect line $y$
Proof that the line $\Omega S$ is limiting parallel to line $y$
In the Poincare Half plane:
Assume $\Omega $ to be the point $(z,0)$
Assume $O$ to be the point $(z,2a)$
Then:
the line $\Omega O$ is the vertical ray $ x=z, y >0$
the hyperbolic line $y$ is the halfcircle through $(z,2a)$ centered at $(z,0)$, This line meets the x axis at $(z \pm 2a,0)$
The horocycle $h$ through $O$ centered around $\Omega$ is the circle $h$ with centre $(z,a)$
Then we need to find the point $S$
Therefor the hyperbolic line $S_xS$ needs to cut the horocycle $h$ where the Euclidean rays $y= t, x= z \pm t , t > 0 $ cut circle h.
So the point $S$ is one of the points $ (z \pm a , a)$
(lets call the point $ (z + a , a) S^+ $ and the point $ (z - a , a) S^- $, one of these points is $S$ (like in your sketch there is also a point $S$ below $O \Omega$)
Then the hyperbolic line $\Omega S$ is one of the the halfcircles centered at $ (z \pm a , 0)$ going trough $\Omega (z, 0) $ The halfcircle $\Omega S$ meets the halfcircle $y$ at $ (z \pm 2a, 0)$ that is on the x axis so the hyperbolic line $\Omega S$ is limiting parallel to the hyperbolic line $y$
This completes the first part of the proof
Proof that if $P_xP < S_xS $ then the line $\Omega P$ will intersect line $y$
The points equidistant to $S_xS$ from $\Omega $ is the hyperbolic hypercycle $e$ and is in the poincare halfplane represented by two Euclidean rays starting at $\Omega $ one going through $S^- $ , the other one going through $S^+ $
Hyperbolic lines $\Omega P$ lines where $P_xP < S_xS $ will cut horocycle $h$ on the arc $S^- O S^+$ They will be represented in the Poincare Half plane model by circles centered on the x axis equidistant from $\Omega$ and $P$, and these circles will cut halfcircle y so these hyperbolic lines $\Omega P$ will cut the hyperbolic line $y$
This completes the second part of the proof
Proof that if $P_xP > S_xS $ then the line $\Omega P$ will intersect line $y$
Hyperbolic lines $\Omega P$ lines where $P_xP > S_xS $ will cut circle $h$ on the arc $S^- \Omega S^+$ they will be represented in the Poincare Half plane model by circles centered on the x axis equidistant from $\Omega$ and $P$, and these circles will not cut the halfcircle y so these hyperbolic lines $\Omega P$ will be ultraparalel to hyperbolic line $y$
This completes the third part of the proof
Now how to translate it to the model you use of the hyperbolic plane .
I have no idea, maybe add a standard proof that the Poincare Half plane model is a represenative of the hyperbolic plane
Can you tell me which book you are using for your study? I only know about two books that use "free flow" hyperbolic planes , but none of them discusses horocycles.
GOOD LUCK