Minimum Shadow Area

An ellipsoid has its first semi-axis of length 15 units along the unit vector $\mathbf{u_1} = \dfrac{1}{9} (8,-4,1)$ , and its second semi-axis of length 30, along the unit vector $\mathbf{u_2} = \dfrac{1}{9} (4, 7, -4)$ , and its third semi-axis of length 10 along the unit vector $\mathbf{u_3} = \dfrac{1}{9} (1, 4, 8 )$ . It is positioned such that it touches the $xy$ plane at the origin (that is, all of the ellipsoid lies above the $xy$ plane, except for one point, the origin). The ellipsoid is subjected to uniform direction light (like sun rays) that produces a shadow of the ellipsoid on the $xy$ plane. By varying the direction of the light rays, find the minimum possible area of that shadow.

Note: In a previous problem it was shown that there is a closed form expression for the shadow area and it is given by

$A = \pi \dfrac{ \sqrt{\mathbf{d_0}^T Q_e \mathbf{d_0} }}{ \sqrt{ | Q_e | } | \mathbf{n}^T \mathbf{d_0}| }$

where $\mathbf{d_0}$ is the unit direction vector of the light rays, $\mathbf{n}$ is the normal to the projection plane, and where the ellipsoid equation is given by $( \mathbf{r} - \mathbf{r_0} )^T Q_e ( \mathbf{r} - \mathbf{r_0} ) = 1$ .