Minimum Distance from an Ellipsoid to a Paraboloid

In the $xyz$ coordinate system, an ellipsoid has its center at the point $(0,0,0)$ , and semi-axes coinciding with the vectors $(1,2,3)$ , $(-5,-2,3)$ , and $(6,-9,4)$ .

A paraboloid has the general shape of the surface $z = x^2 + y^2$ , but its vertex location and orientation are different. The paraboloid's vertex is at the point $(3,7,4)$ , and it opens up in the direction of the vector $(9,-3,-8)$ in the same way in which the surface $z = x^2 + y^2$ opens up in the direction of the positive $z$ axis.

We want to find the minimum distance $d^*$ , between a point $P =(x_1, y_1, z_1 )$ on the ellipsoid and a point $Q = (x_2, y_2, z_2 )$ on the paraboloid. If the minimum occurs with $P^* = ({x_1}^*,{y_1}^*,{z_1}^*)$ , and $Q^* = ( {x_2}^*,{y_2}^*,{z_2}^*)$ , then report the value of $\lfloor 100({x_1}^*+{y_1}^*+{z_1}^* + {x_2}^*+{y_2}^*+{z_2}^* + d^* ) \rfloor$ , where $\lfloor \cdot \rfloor$ is the floor function; for example, $\lfloor 7.6 \rfloor = 7$ .

Inspiration