  In this note, we discuss the possible existence of finite critical
trajectories connecting two zeros a(t) and b(t) of a family of quadratic
differentials satisfying some properties. We treat the cases of holomorphic and
meromorphic quadratic differentials, and we give new proofs concerning the
supports of limit measures of the root-counting measures of the generalized
Laguerre and Jacobi polynomials with varying parameters.
