Maximal ideals of the ring $\frac{K[X,Y]}{\langle XY\rangle}$

Question

Let's assume that we're working with commutative rings with unit and $K$ a field.

I want to compute the maximal ideals of the ring $\frac{K[X,Y]}{\langle XY\rangle}$. I'm given as a hint that every ideal of $K[X,Y]$ is finitely generated?.

So my question is double:

1. Why every ideal of $K[X,Y]$ is finitely generated?.
2. Can you give me a strategy to show what are the maximal ideals of $\frac{K[X,Y]}{\langle XY\rangle}$?

Edit

Reid's page 22 tells what are the maximal ideals of $K[X,Y]$

There is a lemma I think that may be useful, any element of $\frac{K[X,Y]}{\langle XY\rangle}$ can be expressed in a unique way as $k+F_1(X)X+F_2(Y)Y$

Answer 1

Here is an idea for the second question: there exists a natural correspondence between the ideals of $R$ and the ideals of $R/I$, in general: if $p : R \to R/I$ is the natural projection, then $p(J)$ is a (maximal) ideal in $R/I$ if and only if $J$ is a (maximal) ideal in $R$ and $J \supseteq I$. Applying this, the maximal ideals of $K[X,Y] / \langle XY \rangle$ are the images of the maximal ideals of $K[X,Y]$ that contain $\langle XY \rangle$. Again, without knowing more about $K$, it is difficult to tell what are the maximal ideals of $K[X,Y]$.