We introduce two different properties of representations of the free group of rank 2 into the group of hyperbolic isometries PSL(2,C): BQ conditions and primitive stability. We investigate relations between the sets of characters satisfying these conditions, and study certain slices of the character variety separately. The results we get are in line with the conjecture that these two sets actually coincide. We also show that in the diagonal slice of the character variety, namely the set of representations with associated trace triple (z, z, z), z E C, there exists a large class of primitive stable representations which are not discrete and faithful.