**Question: **

(a) |
Part of a landscape generated in a computer game is defined by the quadric surface
\[z=3{{x}^{2}}+2xy+{{y}^{2}}+10x+2y+5\]. |
|||

(i) | Obtain the critical point of the surface and determine if it is a maximum, a minimum or a saddle point. | |||

(ii) | Determine the equation of the tangent plane to the surface at the point where \[x=-1\] and \[y=0\]. | |||

(iii) |
Find or state a vector that is parallel to the normal to the surface where \[x=-1\] and \[y=0\]. |
[11] | ||

(b) | (i) | Construct the bilinear surface patch based on the line segments joining \[A\left( 1,\,1,\,0 \right)\] with \[B\left( 2,\,1,\,1 \right)\] and \[C\left( 1,\,2,\,0 \right)\] with \[D\left( 2,\,2,\,0 \right)\]. | ||

(ii) |
Obtain the parametric equations of the curve joining A to D. |
|||

(iii) |
Show that the curve joining B to C is a parabola lying in the plane \[x+y=3\]. |
[9] |