Prime ideals in $k[x,y]/(xy-1)$. Let $k$ a field. Let $f$ be the ring injective homomorphism
$$ f:k[x] \rightarrow k[x,y]/(xy-1)$$
obtained as the composition of the inclusion $k[x] \subset k[x,y]$ and the natural projection map $ f:k[x,y] \rightarrow k[x,y]/(xy-1)$.

Prove that there isn't any prime ideal in the ring $k[x,y]/(xy-1)$ whose contraction in $k[x]$ is the prime ideal $(x)$.
Is there any prime ideal in $k[x,y]/(xy-1)$ whose contraction is $(x-1)$?

Thanks! :)

HINT:
$k[x,y]/(xy-1)$ is naturally isomorphic to the ring of fractions $k[x][\frac{1}{x}] = S^{-1}\ k[x]$, where $S= \{1,x,x^2, \ldots\}$. 
