Reflection across an Ellipsoid

A reflective ellipsoid is given by

$\dfrac{x^2}{16} + \dfrac{y^2}{9} + \dfrac{(z - 2)^2}{4} = 1$

A light source is placed at point $A (10, 4, 3)$ . Find the point $C(x, y, z)$ on the surface of the ellipsoid such that a light ray radiating from $A$ reflects off the surface of the ellipsoid at $C$ , and the reflected ray passes through point $B (8, 12, 7)$ . Find the sum of the coordinates of point $C$ , that is, find $x + y + z$ .

Details and Assumptions:

• This problem is equivalent to finding the point $C$ on the surface of the ellipsoid that will minimize the sum of distances $AC + CB$ .