We're doing gradient vectors, and I'm kind of confused by the applications of them.

Here's a question where I'm not entirely sure where to start:

Find the points on the ellipsoid where the tangent plane is parallel to the plane

First, I know that the normal plane to the ellipsoid at these points is going to be <3,-1,3>. I know that to find the gradient vector I take the partials with respect to x and y, but then I'm not sure where to go from there. I don't have points I'm working with, I'm being asked to find points. The whole directional derivative thing is actually very confusing in general--if someone could help me out that would be great!