Since you suggest this is homework, I'll put this in the form of hints:
(1) Remember that the set of associated primes for a primary decomposition is unique. So you are looking for $\mathfrak{p} \cap \mathfrak{q} \cap \mathfrak{r}$ where $\sqrt{\mathfrak{p}} = (x)$, $\sqrt{\mathfrak{p}} = (y)$ and $\sqrt{\mathfrak{r}} = (x,y)$.
(2) Show that $\mathfrak{p}$ and $\mathfrak{q}$ must be $(x)$ and $(y)$. So the place where you have room to play is in choosing $\mathfrak{r}$.
At this point, I have trouble giving good general hints. The next two, which are far more helpful, will be put in ROT13.
(3) Lbh jvyy abg or noyr gb fbyir guvf ceboyrz vs lbh fgvpx gb vqrnyf trarengrq ol zbabzvnyf.
(4) Gur rknzcyr V sbhaq jnf nyfb trarengrq va qrterr gjb.