Question 5 - A-Level Maths

The rectangular hyperbola $H$ has equation $x y = 36$
1. Use calculus to show that the equation of the tangent to $H$ at the point $P \left( 6 t , \frac { 6 } { t } \right)$ is
$$y t ^ { 2 } + x = 12 t$$ The point $Q \left( 12 t , \frac { 3 } { t } \right)$ also lies on $H$.
Find the equation of the tangent to $H$ at the point $Q$. The tangent at $P$ and the tangent at $Q$ meet at the point $R$.
Show that as $t$ varies the locus of $R$ is also a rectangular hyperbola.