Let $F=F(x,y),G=G(x,y) \in \mathbb{C}[x,y]$, $u=u(x),v=v(x) \in \mathbb{C}[x]$, $c \in \mathbb{C}$.

Denote:

$A=F(x,y)y+u(x)(x-c)=Fy+u(x-c)$,

$B=G(x,y)y+v(x)(x-c)=Gy+v(x-c)$.

When $I_{A,B}:=\langle A,B \rangle$ is a maximal ideal of $\mathbb{C}[x,y]$?

It is well-known that maximal ideals of $\mathbb{C}[x,y]$ are of the form: $\langle x-a,y-b \rangle$, $a,b \in \mathbb{C}$, see this question. Observe that, for example, $\langle x-a, y-b+(x-a)^5 \rangle$ is maximal, since $\langle x-a, y-b+(x-a)^5 \rangle = \langle x-a,y-b \rangle$.

**Remarks:**
1. Notice that $I_{A,B} \subseteq \langle x-c, y \rangle$.

- What if we further assume that $\gcd(u,v)=\gcd(u(x),v(x))=1$?

Any hints and comments are welcome!