Reflection across a Paraboloid

A reflective paraboloid is given by

$z = 0.25 x^2 + 0.1 y^2$

A light source is placed at point $A (10,4,3)$ . A light ray radiating from point $A$ is reflected by the paraboloid at a point $C (x, y, z)$ , and the reflected ray is to pass through the point $B(8,12,7)$ . Find the coordinates of point $C$ , and enter the sum $x + y + z$ .

Details and Assumptions:

This problem is equivalent to finding the point $C$ on the surface of the paraboloid that minimizes the sum of distances $AC + CB$ .