Expand ellipsoid to contain a ray

Hi,

Given an ellipsoid A arbitrarily oriented in 3D with center at the
origin, I would like to stretch/expand it to an ellipsoid B with the
same center, such that B fully contains a given vector v (ray from the
origin). There are two conditions: that A be completely within B, and
that B be as small as possible. Note that B's orientation can be
different from A's.
The second condition exists because otherwise the easiest thing would
be to scale A equally in all directions to contain v, but that
penalizes directions v is not in.

I have the symmetric matrix that defines the ellipsoid and also the
vector that defines the ray. How do I go about this? Any help would be
immensely appreciated. Thank you.

Khan

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