Problem

The rectangular hyperbola $H$ has equation $xy = 100$

The point $P\left(10t, \frac{10}{t}\right)$, where $t \neq 0$, lies on $H$.

(a) Use calculus to show that the normal to $H$ at $P$ has equation

$t^3x - ty = 10(t^4 - 1)$

The normal to $H$ at $P$ meets the $y$-axis at the point $Q$.

Given that the area of triangle $OPQ$, where $O$ is the origin, is $750$

(b) determine both possible pairs of coordinates of the point $P$.