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

For any non-uniform lattice $\mathrm{\Gamma }$ in $SL(2,R)$, we describe the limit distribution of orthogonal translates of a divergent geodesic in $\Gamma \backslash SL(2,R)$\mathrm{.}Asanapplication,foraquadraticform$Q$ofsignature$(2,1)$,alattice$\Gamma $initsisometrygroup,and$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 $v_0\mathrm{\Gamma }$ of norm at most $T$, when the stabilizer of $v_0$ in $\mathrm{\Gamma }$ 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 $c\cdot T\log T+O(T)$.