- The parabola \(C\) has equation \(y ^ { 2 } = 36 x\)
The point \(P \left( 9 t ^ { 2 } , 18 t \right)\), where \(t \neq 0\), lies on \(C\)
- Use calculus to show that the normal to \(C\) at \(P\) has equation
$$y + t x = 9 t ^ { 3 } + 18 t$$
- Hence find the equations of the two normals to \(C\) which pass through the point (54, 0), giving your answers in the form \(y = p x + q\) where \(p\) and \(q\) are constants to be determined.
Given that
- the normals found in part (b) intersect the directrix of \(C\) at the points \(A\) and \(B\)
- the point \(F\) is the focus of \(C\)
- determine the area of triangle \(A F B\)