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  \item The ellipse $E$ has equation
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$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$

(a) Determine the exact value of the eccentricity of $E$

The points $P ( 4 \cos \theta , 3 \sin \theta )$ and $Q ( 4 \cos \theta , - 3 \sin \theta )$ lie on $E$ where $0 < \theta < \frac { \pi } { 2 }$ The line $l _ { 1 }$ is the normal to $E$ at the point $P$

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(b) Use calculus to show that $l _ { 1 }$ has equation

$$4 x \sin \theta - 3 y \cos \theta = 7 \sin \theta \cos \theta$$

The line $l _ { 2 }$ passes through the origin and the point $Q$ The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $R$\\
(c) Determine, in simplest form, the coordinates of $R$\\
(d) Hence show that, as $\theta$ varies, $R$ lies on an ellipse which has the same eccentricity as ellipse $E$

\hfill \mbox{\textit{Edexcel FP1 2023 Q4}}