Let $(x_1,\dots,x_n) \in \mathbb{C}^n$ be the set of points $S$, where the polynomial $P(x_1,\dots,x_n,t)=0$ has (at least) two roots $t$ with same magnitude.

Is it true (in general), that the discriminant $D(x_1,\dots,x_n) = Discr_t(P)$ satisfy $D(x_1,\dots,x_n)=0$ on the boundary of $S$?

What happens if we restrict $S$ to be the set where the two roots of $P$ with largest magnitude must have the same magnitude?