The free energy of membrane deformation when a tilted ion channel is imbedded in a membrane is analyzed using a well-known continuum model. The ion channel is assumed to be closed when tilted and open when untilted. When the energetics of the membrane deformation is analyzed with the continuum model, explicit boundary conditions are usually adopted at a contact position between the channel and the membrane and at a position far from the channel. However, it is difficult to determine the contact position when the channel has a tilted orientation. Depending on the tilt angle, the difference in the contact position between the upper and the lower parts of the membrane could be as large as the thickness of the membrane. Hence, in this work, the dependence of the free energy of membrane deformation on the contact position is investigated in detail. Analysis shows that the contact position significantly affects the energetics of the ion channel gating when the energy cost for a thickness change of the membrane is not negligible. The analysis also shows that the energetics of rotational gating mechanisms, as discussed in a previous article using a decoupled model, can be reproduced for some reasonable values of the contact position.