  We establish limit laws for the distribution in small intervals of the roots
of the quadratic congruence $\mu^2 \equiv D \bmod m$, with $D > 0$ square-free
and $D\not\equiv 1 \bmod 4$. This is achieved by translating the problem to
convergence of certain geodesic random line processes in the hyperbolic plane.
This geometric interpretation allows us in particular to derive an explicit
expression for the pair correlation density of the roots.
