If the position of some charge $Q$ is known, the boundary condition is $u=0$ on some parabolic surface, and we know the image charge has its electric volume of $Q'$, then how can I determine the position of the image charge?
Same questions goes for the hyperbolic curve boundary: how can I determine the position of $Q'$?
I think may be there is a way to transform the coordinates to make everything into an easily-handled form, but I am not sure about it. Another solution I thought about is to put this question into a general question based on basic Poisson's equation and Laplace equation, but I do not have a specific idea on how to conduct this.