Problem

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 $P$.

(a) Show that an equation for $l$ is

$$5x\sin\theta - 4y\cos\theta = 9\sin\theta\cos\theta.$$

The point $F$ is the focus of $E$ that lies on the positive $x$-axis.

(b) Determine the coordinates of $F$.

The line $l$ crosses the $x$-axis at the point $Q$.

(c) Show that

$$\frac{QF}{PF} = e$$

where $e$ is the eccentricity of $E$.