Limits of translates of divergent geodesics and Integral points on one-sheeted hyperboloids

For any non-uniform lattice $\Gamma$ in $SL(2,R)$, we describe the limit distribution of orthogonal translates of a divergent geodesic in $\Gamma \backslash SL(2,R)$. As an application, for a quadratic form $Q$ of signature $(2,1)$, a lattice $\Gamma$ in its isometry group, and $v_0\in R^3$ with $Q(v_0)>0$, we compute the asymptotic (with a logarithmic error term) of the number of points in a discrete orbit $v_0\Gamma$ of norm at most $T$, when the stabilizer of $v_0$ in $\Gamma$ is finite. Our result in particular implies that for any non-zero integer $d$, the smoothed count for number of integral binary quadratic forms with discriminant $d^2$ and with coefficients bounded by $T$ is asymptotic to $c\cdot T \log T +O(T)$.