I recently learnt that the maximal ideals of $\mathbb C[x, y]$ are precisely the ones of the form $(x-a, y-b)$ for some $a, b\in \mathbb C$.

I am unable to prove it.

So I considered an easier version of the problem.

Let $M=(p, q)$ be an ideal in $\mathbb C[x, y]$, where $p$ and $q$ are elements of $\mathbb C[x, y]$. If $M$ is maximal, then $\deg p=\deg q=1$.

I am stuck even at this.

An ideal is maximal if quotienting the parent ring with it gives a field, thus $\mathbb C[x, y]/(p, q)$ is a field. But I am having no ideas.