Tangent plane parallel to given direction

2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).

1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
2. Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
3. Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
4. Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.

5 A surface \(S\) is defined for \(z \geqslant 0\) by \(x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 126\). \(C\) is the set of points on \(S\) for which the tangent plane to \(S\) at that point intersects the \(x - y\) plane at an angle of \(\frac { 1 } { 3 } \pi\) radians. Show that \(C\) lies in a plane, \(\Pi\), whose equation should be determined.