  Let (RU_1, R U_2) be a given bivariate scale mixture random vector, with R>0
being independent of the bivariate random vector (U_1,U_2). In this paper we
derive exact asymptotic expansions of the tail probability P{RU_1> x, RU_2>
ax}, a \in (0,1] as x tends infintiy assuming that R has distribution function
in the Gumbel max-domain of attraction and (U_1,U_2) has a specific tail
behaviour around some absorbing point. As a special case of our results we
retrieve the exact asymptotic behaviour of bivariate polar random vectors. We
apply our results to investigate the asymptotic independence and the asymptotic
behaviour of conditional excess for bivariate scale mixture distributions.
