1 A surface, \(S\), is defined in 3-D by \(z = f ( x , y )\) where \(f ( x , y ) = 12 x - 30 y + 6 x y\).
- Determine the coordinates of any stationary points on the surface.
- The equation \(\mathrm { z } = \mathrm { f } ( \mathrm { x } , \mathrm { a } )\), where \(a\) is a constant, defines a section of S .
Given that this equation is \(\mathrm { z } = 24 \mathrm { x } + \mathrm { b }\), find the value of \(a\) and the value of \(b\).
The diagram shows the contour \(z = 12\) and its associated asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{33c9e321-6044-45c4-bf37-0a6da3ecaf0d-2_860_1143_742_242} - Find the equations of the asymptotes.
- By forming grad \(g\), where \(g ( x , y , z ) = f ( x , y ) - z\), find the equation of the tangent plane to \(S\) at the point where \(x = 3\) and \(y = 2\). Give your answer in vector form.
The point \(( 0,4 , - 120 )\), which lies on S , is denoted by A .
The plane with equation \(\mathbf { r }\). \(\left( \begin{array} { r } 3
3
- 2 \end{array} \right) = 52\) is denoted by \(\Pi\). - Show that the normal to S at A intersects \(\Pi\) at the point \(( - 360,304 , - 110 )\).