5.
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\caption{Figure 2}
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Diagram not drawn to scale
$$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$
The parabola C has equation \(\mathrm { y } ^ { 2 } = 16 \mathrm { x }\)
- Deduce that the point \(\mathrm { P } \left( 4 \mathrm { p } ^ { 2 } , 8 \mathrm { p } \right)\) is a general point on C .
The line I is the tangent to C at the point P .
- Show that an equation for I is
$$p y = x + 4 p ^ { 2 }$$
The finite region R , shown shaded in Figure 2, is bounded by the line I , the x -axis and the parabola C.
The line \(I\) intersects the directrix of \(C\) at the point \(B\), where the \(y\) coordinate of \(B\) is \(\frac { 10 } { 3 }\) Given that \(\mathrm { p } > 0\) - show that the area of R is 36
\section*{Q uestion 5 continued}