What is at what point on the ellipsoid is the tangent plane parallel?

to a given plane.

At any point on an ellipsoid, there are two tangent planes that pass through that point. The tangent planes are defined as the planes that touch the surface of the ellipsoid at that point and have a normal vector that is perpendicular to the surface of the ellipsoid at that point.

A given plane is parallel to a tangent plane of the ellipsoid at a particular point if and only if the normal vector of the given plane is parallel to the normal vector of the tangent plane at that point.

Therefore, to find the point on the ellipsoid where the tangent plane is parallel to a given plane, you need to find the normal vector of the given plane and then find the point on the ellipsoid where the normal vector of the tangent plane is parallel to the normal vector of the given plane.

This can be done using calculus, by finding the gradient of the ellipsoid equation and the equation of the given plane and then solving for the point of intersection that satisfies the condition of parallel normal vectors.