  Cherenkov radiation (CR) generated by a charge moving through a hollow
conical target made of dielectric material is analyzed. We consider two cases:
the charge moves from the base of the cone to its top (``straight'' cone) or
from the top to the base (``inverted'' cone). Unlike previous papers, a nonzero
shift of the charge trajectory from the symmetry axis is taken into account
which leads to generation of asymmetric CR. The most interesting effect is the
phenomenon of ``Cherenkov spotlight'' which has been reported earlier for
axially symmetric problems. This effect allows essential enhancement of the CR
intensity in the far-field region by proper selection of the target's
parameters and charge velocity. Here we describe the influence of charge shift
on CR far-field patterns paying the main attention to the ``Cherenkov
spotlight'' regime. Influence of variation of the charge speed on this
phenomenon is also investigated.
