Problem

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The rectangular hyperbola $H$ has equation $xy = 36$

The point $P(4, 9)$ lies on $H$

(a) Show, using calculus, that the normal to $H$ at $P$ has equation

$4x - 9y + 65 = 0$

The normal to $H$ at $P$ crosses $H$ again at the point $Q$

(b) Determine an equation for the tangent to $H$ at $Q$, giving your answer in the form $y = mx + c$ where $m$ and $c$ are rational constants.