Ok, I'm with you now, I think.
Ellipse = 2 dimensional shape in a flat plane
Ellipsoidal = 3 dimensional shape (a concave ellipse, yes?) As per picture.
Right! SUB of Gregory type antenna has shape of the cut of rotatory ellipsoid (shape similar to melon). Of course, our "melon" should be at right size, dimensions and with properly made cut.
So you want us to determine the actual final position of F2 so that when drawn using F1 & F2 as the two fixed pins for the formation of the ellipse, the edges of the ellipse will pass through points A & B, is that correct?
Yes that what we will be doing.
So to determine where the final position of F2 is we will have to adjust it's position along the axis x-y, as per the diagram, is that correct?
Will not limit position of F2 by moving it only along the X-Y axe, it's position could be anywhere.
But more important for everybody, at this moment, is to find the way how to draw ellipse. It could be easily done on paper or electronically, you chose your own way. For example like this
For example, here I drew ellipse from focuses F1 and F2 and outside line of ellipse goes via point A
But, if I move F2 to another place, I can draw ellipse from focuses F1 and F2 and A and B are also connected by line