2 A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), where \(\mathrm { g } ( x , y , z ) = x ^ { 2 } + 3 y ^ { 2 } + 2 z ^ { 2 } + 2 y z + 6 x z - 4 x y - 24\). \(\mathrm { P } ( 2,6 , - 2 )\) is a point on the surface \(S\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
2. Find the equation of the normal line to the surface \(S\) at the point P .
3. The point Q is on this normal line and close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = h\), where \(h\) is small. Find, in terms of \(h\), the approximate perpendicular distance from Q to the surface \(S\).
4. Find the coordinates of the two points on the surface at which the normal line is parallel to the \(y\)-axis.
5. Given that \(10 x - y + 2 z = 6\) is the equation of a tangent plane to the surface \(S\), find the coordinates of the point of contact.