Do the set $\{ p(x,y): P(a,b)=0\}$ is a maximal( or prime) ideal of $\mathbb{C}[x,y]$. If so what will be its principal ideal form representation.(Since all ideals are principal).
I feel the ideal is maximal as viewing $\mathbb{C}[x,y]$ as a $R[x]$ and $\{ p(x,y): P(a,b)=0\}$ as a $<x-\alpha>$, (where, $\alpha$ corresponds to (a,b))
Hint. To prove that your ideal is maximal, apply the first isomorphism theorem to $P\in\mathbb{C}[x,y]\mapsto P(a,b)\in\mathbb{C}$.
If you want to go further, you can prove that your ideal is $(x-a,y-b)$ (use generalized long division of polynomials carefully), and that it is NOT a principal ideal (if so, a generator is a common divisor of $x-a$ and $y-b$, so...)