7 The 'parabolic' TV satellite dish in the diagram can be modelled by the surface generated by the rotation of part of a parabola around a vertical \(z\)-axis. The model is represented by part of the surface with equation \(z = \mathrm { f } ( x , y )\) and \(O\) is on the surface. The point \(P\) is on the rim of the dish and directly above the \(x\)-axis.
The object, \(B\), modelled as a point on the \(z\)-axis is the receiving box which collects the TV signals reflected by the dish.
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1. The horizontal plane \(\Pi _ { 1 }\), containing the point \(P\), intersects the surface of the model in a contour of the surface.
   (a) Sketch this contour in the Printed Answer Booklet.
   (b) State a suitable equation for this contour.
2. A second plane, \(\Pi _ { 2 }\), containing both \(P\) and the \(z\)-axis, intersects the surface of the model in a section of the surface.
   (a) Sketch this section in the Printed Answer Booklet.
   (b) State a suitable equation for this section.
3. A proposed equation for the surface is \(z = a x ^ { 2 } + b y ^ { 2 }\). What can you say about the constants \(a\) and \(b\) within this equation? Justify your answers.
4. The real TV satellite dish has the following measurements (in metres): the height of \(P\) above \(O\) is 0.065 and the perimeter of the rim is 2.652 . Using this information, calculate correct to three decimal places the values of
   • \(a\) and \(b\),
   • any other constants stated within the answers to parts (i)(b) and (ii)(b).
   • Incoming satellite signals arrive at the dish in linear "beams" travelling parallel to the \(z\)-axis. They are then 'bounced' off the dish to the receiving box at \(B\).
   • On the diagram for part (ii)(a) in the Printed Answer Booklet draw some of these beams and mark \(B\).
   • If the values of \(a\) and \(b\) were changed, what would happen?