  In this paper, we introduce a two-parameters determinantal point process in
the Poincar\'e disc and compute the asymptotics of the variance of its number
of particles inside a disc centered at the origin and of radius $r$ as $r$
tends to $1$. Our computations rely on simple geometrical arguments whose
analogues in the Euclidean setting provide a shorter proof of Shirai's result
for the Ginibre-type point process. In the special instance corresponding to
the weighted Bergman kernel, we mimic the computations of Peres and Virag in
order to describe the distribution of the number of particles inside the disc.
