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

For any non-uniform lattice Γ in SL₂(?), we describe the limit distribution of orthogonal translates of a divergent geodesic in Γ\SL₂(?). As an application, for a quadratic form Q of signature (2, 1), a lattice Γ in its isometry group, and v ₀ ∈ ?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 ₀Γ of norm at most T, when the stabilizer of v ₀ in Γ 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).