  Motivated by the study of the asymptotic behavior of Jacobi polynomials
$\left( P_{n}^{(nA,nB)}\right) _{n}$ with $A\in \mathbb C$ and $B>0$ we
establish the global structure of trajectories of the related rational
quadratic differential on $\mathbb C$. As a consequence, the asymptotic zero
distribution (limit of the root-counting measures of $\left(
P_{n}^{(nA,nB)}\right) _{n}$) is described. The support of this measure is
formed by an open arc in the complex plan (critical trajectory of the
aforementioned quadratic differential) that can be characterized by the
symmetry property of its equilibrium measure in a certain external field.
