Maximal and prime ideals in polynomial rings

Question

Consider the ideal $I=(ux,uy,vx,uv)$ in the polynomial ring $\mathbb{Q}[u,v,x,y]$, where $u,v,x,y$ are indeterminates. Prove that every prime ideal $P$, containing $I$ contains the ideal $A=(x,y)$ or the ideal $B=(u,v)$. It is also to be proved that the following 3 statements are false: (a) Every prime ideal containing I contains the ideal $(x,y)$. (b) Every maximal ideal containing $I$ contains the ideal $(u,v)$. (c) Every maximal ideal containing I contains the ideal $(u,v,x,y)$. Towards solving this problem, I have that $\mathbb{Q}[x,y,u,v]\setminus P$ is an integral domain since $P$ is prime. I am not able to link this fact with $P$ containing $A$ or $B$. Also I just can’t get counter examples to prove that statements (a), (b) and (c) are false. I will be thankful for help for solving this problem.