  We study the Hankel determinant of the generalized Jacobi weight
$(x-t)^{\gamma}x^\alpha(1-x)^\beta$ for $x\in[0,1]$ with $\alpha, \beta>0$, $t
< 0 $ and $\gamma\in\mathbb{R}$. Based on the ladder operators for the
corresponding monic orthogonal polynomials $P_n(x)$, it is shown that the
logarithmic derivative of Hankel determinant is characterized by a
$\tau$-function for the Painlev\'e VI system.
