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 similar to In this paper we investigate the asymptotic behaviour of the conditional survivor probability (I) over bar rho, u( y) := P {rho S(1) + root 1 - rho(2)S(2) > y|S(1) > u}, rho is an element of (- 1, 1), is an element of IR when u approaches the upper endpoint of F. On the density function of similar to we impose 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 I., u, which is then utilised to construct two estimators for the conditional distribution function 1 - (I) over bar rho,u. Furthermore, we allow Theta to depend on u.