5. The hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$
The point \(P ( 4 \sec \theta , 3 \tan \theta ) , 0 < \theta < \frac { \pi } { 2 }\), lies on \(H\).
- Show that an equation of the normal to \(H\) at the point \(P\) is
$$3 y + 4 x \sin \theta = 25 \tan \theta$$
The line \(l\) is the directrix of \(H\) for which \(x > 0\)
The normal to \(H\) at \(P\) crosses the line \(l\) at the point \(Q\). Given that \(\theta = \frac { \pi } { 4 }\) - find the \(y\) coordinate of \(Q\), giving your answer in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are rational numbers to be found.
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