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

For any non-uniform lattice Gamma in SL2(R), we describe the limit distribution of orthogonal translates of a divergent geodesic in Gamma\SL2(R). As an application, for a quadratic form Q of signature (2, 1), a lattice Gamma in its isometry group, and v(0) is an element of 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 the number of integral binary quadratic forms with discriminant d (2) and with coefficients bounded by T is asymptotic to c . T log T + O(T).