Warning: I am not a mathematician, so excuse me if the problem at hand seems trivial or incomplete.
So, I am working on the following problem: I have to find the smallest (i.e smallest volume) N-dimensional ellipsoid that contains an existing $N$-dimensional ellipsoid and a given point. (This point will lie on the surface of the new ellipsoid.)
The known ellipsoid is described by the general equation:
$$ (x-x_0)^TM_1(x-x_0) \le 1 $$
The unknown ellipsoid would similarly be: $$ (x-x_0)^TM_2(x-x_0) \le 1 $$
and would completely contain the ellipse given by $M_1$ and a point $x_{new}$.
So the following should be true:
$$ (x_{new}-x_0)^TM_2(x_{new}-x_0) = 1 $$
and gives a first equation to solve the problem. Now however I do not know how to use the other information I have (i.e. that the second ellipsoid should completely contain the first ellipsoid and be the smallest ellipsoid with this property).
I have done some reading and have not found a way to solve this. I suppose one could treat this as a "smallest ellipsoid containing a set of points" problem and solve it this way. But I wondered if there is an analytical way to solve this. It seems there should be, but I am a little rusty in this topic so could not come to an answer myself.
Here a visual representation of the problem:
