Fiber is not domain for $a\neq 0$

Question

Let $S=k[x,t]\times k[t,t^{-1}]$ as algebra over $k[t]$. Fiber $S/(t-a)S$ over $(t-a)\subset k[t]$ is not domain for $a\neq 0$.

I failed to find the example to see this is not domain when $a\neq 0$. If $a=0$, $S/(t-a)S=k[x]$ clearly a domain.