Question 8 - A-Level Maths

$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$

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 $$x b \cosh \theta - y a \sinh \theta = a b$$ 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 $P$. 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 $Q$.
Show that, as $\theta$ varies, the locus of the mid-point of $P Q$ has equation $$x \left( 4 y ^ { 2 } + b ^ { 2 } \right) = a b ^ { 2 }$$