Probability density function good example Even though a fast-food chain might advertise a hamburger as weighing a quarter-pound, you can well imagine that it is not exactly 0.25 pounds. One randomly selected hamburger might weigh 0.23 pounds while another might weigh 0.27 pounds.  What is the probability that a randomly selected hamburger weighs between 0.20 and 0.30 pounds? That is, if we let X denote the weight of a randomly selected quarter-pound hamburger in pounds, what is P(0.20 < X < 0.30)?
 A: This is an absolute statement:

If a line is perpendicular to a circle (it is perpendicular to the tangents through its intersection points) then that line will go through the center of the circle. This statement does not depend on the postulate of parallelism.

So, the statement does not have anything to do with the Poincaré disk or half plane model. The statement will be true in any model of the hyperbolic or the Euclidean plane.
Having understood this, one may ask questions about the different intuitions of the same statement in different models of the hyperbolic plane. 
For instance, in the Klein model, this is how to intuit the statement

Here, the green curve is the edge of the Klein model. The black line is perpendicular to the red tangents, that is to the circle.
Below, our absolute statement is shown in the usual model of the Euclidean geometry:

Finally, this is the visualization in the Poincaré disk model

The misleading specialty of the Poincare models (half plane and disk) is that the circles look Euclidean circles...
But, behind all these pictures there is a very simple absolute statement quoted above.
A: It can be ordered in a wider context using bipolar coordinates in the complex plane for hyperbolic geodesics, where  $ \tau = const $ and $\sigma =const $ orthogonal circles being real and imaginary parts of a single complex variable 
$$ w= \tau +i \sigma\,; \,z= a \coth(w/2)\,; $$
inside the " Poincaré compounded " model. I present here a comprehensive/integrated  picture of the two Poincaré models as follows :
Under this re-arrangement we consider a hyperbolic geodesic line $L$ in the Poincaré plane model  and let $C$ be a euclidean circle set contained in the hyperbolic plane.  We can see that $L$ is perpendicular to euclidean circle set $C$ invariably (if and only if) $L$ passes through the hyperbolic centre of $C$, viz., center of concurrency $O$. The euclidean circles are non-concentric,  from centre offset retaining a constant tangent length from the fixed central pole on real axis. 
The point of concurrecy is a singular solution for set of circles $L,C$. In this sense they have been already termed as "foci" in Wikipaedia cited.
It is seen that this compounded model addresses the given question in title by integrating the Poincaré plane & disc models. In either case red $L$ circles are  hyperbolic geodesics and interstingly placed blue euclidean $C$ circles to pass through a single focal point as hyperbolic center $O$ and its conjugate with powers $\pm a^2$.If $O$ has coordinates $(0,a)$ then the $L$ circle set has power $- a^2$ and likewise the $C$ circles set has power $+a^2, $ corrosponding to $\tau \rightarrow \infty. $
Compound "Poincaré disk + plane" models
Wiki Bipolar Cordinates

