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It's clear that if the line charge is located in the center, the surface charge density is a constant because all points of the inner surface of the cavity are equally close to the line charge.

So when the line charge is off-center, the surface charge distribution has to be varying around the center with the angle. Yet, the inner surface of the cavity still has to be a equipotential surface.

Can anyone help me with an idea of how to solve this problem? I will for sure need the cosine law to determine the distance of the surface of the cavity from the line charge, but from there...?

Thank you very much for your help!

Regards, John