We work over a field $k$. Let $B$ be an algebraic variety over $k$. Suppose we are given a family of subvarieties of $\textbf P_k^n$ with base $B$, by which I mean a subvariety $V\subset B\times\textbf P_k^n$ together with the projection $\pi:V\to B$. The members of the family are the fibers $V_b\subset\textbf P_k^n$. Suppose we also have a subvariety $X\subset\textbf P_k^n$.
$\textbf{Question}\,\,1$: How to show that the locus $L=\{b\in B\,\,|\,\,V_b\cap X\neq \emptyset \}$ is closed in $B$?
I just observed that $\pi$ is a closed map (it is proper), and noticed that $L=\pi(Z)$ with $Z=\{(b,x)\in B\times X\,\,|\,\,x\in X\cap V_b\}$. But how to show then that $Z$ is closed in $B\times X$?
$\textbf{Question}\,\,2.$ Also, (forgetting about $X$) if we are given a second family $\pi':W\to C$, I would like to see that the locus $L'=\{(b,c)\,\,|\,\,V_b\cap W_c\neq\emptyset\}$ is closed in $B\times C$.
My problem here - and also above - is that I cannot translate in a nice way the nonempty condition, which is the only one I have. Moreover, if I try to make $L,L'$ explicit, I just come up with infinite unions of closed, while I would like intersections, for instance.
Thank you for any advise.
[Note: these problems come from Joe Harris, "Algebraic Geometry. A first course."]