  In this article, we point out the connections between the distinguished
varieties introduced by Agler and McCarthy with certain uniform algebras on
bidisc studied by Samuelsson and Wold. We also prove analogues of
Samuelsson-Wold result for the domains in $\mathbb{C}^2$ that are the images of
the bidisc under certain proper polynomial map on $\mathbb{C}^2$. We also give
a description of polynomial convex hull of graph of anti-holomorphic polynomial
over the distinguished boundary of such domains. We mention the case for the
symmetrized bidisc as an example.
