  Let (S_1,S_2)=(R \cos(\Theta), R \sin (\Theta)) be a bivariate random vector
with associated random radius R which has distribution function $F$ being
further independent of the random angle \Theta. In this paper we investigate
the asymptotic behaviour of the conditional survivor probability
\Psi_{\rho,u}(y):=\pk{\rho S_1+ \sqrt{1- \rho^2} S_2> y \lvert S_1> u}, \rho
\in (-1,1),\in R when u approaches the upper endpoint of F. On the density
function of \Theta we require a certain local asymptotic behaviour at 0,
whereas for F we require that it belongs to the Gumbel max-domain of
attraction. The main result of this contribution is an asymptotic expansion of
\Psi_{\rho,u}, which is then utilised to construct two estimators for the
conditional distribution function 1- \Psi_{\rho,u}. Further, we allow \Theta to
depend on u.
