Find the coordinates of ellipsoid located below the surface

I would like to consider only the part of the ellipsoid located under the yellow surface (it should be attached to the surface which means i also need the coordinates on the yellow surface as well as the below). How can I find that area so that everytime I have an new ellipsoid, the only the below part of it would be plotted rather than the entire of it.

[Tarunbir Gambhir]

Do you have the equations for the yellow surface and ellipsoid?