9. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$$
The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 4 \sin \theta )\) where \(0 < \theta < \frac { \pi } { 2 }\)
The line \(l\) is the normal to the ellipse at the point \(P\).
- Show that an equation for \(l\) is
$$5 x \sin \theta - 4 y \cos \theta = 9 \sin \theta \cos \theta$$
The point \(F\) is the focus of \(E\) that lies on the positive \(x\)-axis.
- Determine the coordinates of \(F\).
The line \(l\) crosses the \(x\)-axis at the point \(Q\).
- Show that
$$\frac { | Q F | } { | P F | } = e$$
where \(e\) is the eccentricity of \(E\).