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 targets 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.
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