Hyperbola tangent/normal equation derivation

A question is this type if and only if it asks to derive the equation of a tangent or normal to a hyperbola at a parametric point using calculus.

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The hyperbola $H$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the ellipse $E$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ where $a > b > 0$ The line $l$ is a tangent to hyperbola $H$ at the point $P ( a \sec \theta , b \tan \theta )$, where $0 < \theta < \frac { \pi } { 2 }$
1. Using calculus, show that an equation for $l$ is
$$b x \sec \theta - a y \tan \theta = a b$$ Given that the point $F$ is the focus of ellipse $E$ for which $x > 0$ and that the line $l$ passes through $F$,
show that $l$ is parallel to the line $y = x$

Use differentiation to show that an equation of the tangent to H at P is $$3 x = 5 y \sin u + 15 \cos u$$
Prove that P is the mid-point of RS.

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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Use calculus to show that the equation of the tangent to $H$ at the point $(a\cosh\theta, b\sinh\theta)$ may be written in the form $$xb\cosh\theta - ya\sinh\theta = ab$$ [4] The line $l_1$ is the tangent to $H$ at the point $(a\cosh\theta, b\sinh\theta)$, $\theta \neq 0$. Given that $l_1$ meets the $x$-axis at the point $P$,
find, in terms of $a$ and $\theta$, the coordinates of $P$. [2] The line $l_2$ is the tangent to $H$ at the point $(a, 0)$. Given that $l_1$ and $l_2$ meet at the point $Q$,
find, in terms of $a$, $b$ and $\theta$, the coordinates of $Q$. [2]
Show that, as $\theta$ varies, the locus of the mid-point of $PQ$ has equation $$x(4y^2 + b^2) = ab^2$$ [6]

Show that an equation of the normal to $C$ at the point $P(a \sec t, b \tan t)$ is $$ax \sin t + by = (a^2 + b^2) \tan t.$$ [6]