Recover a hyperbola from tangent-intercept midpoints

The curve $C$ has $x=3/t$, $y=2t$.

(i) The line $y=2x$ cuts $C$ at $A$ and $B$. Find the exact length of $AB$.3 marks

(ii) The tangent at $P=(3/p,2p)$ meets the x-axis at $D$ and the y-axis at $E$. Let $F$ be the midpoint of $DE$. Find the cartesian equation of the curve traced by $F$ as $p$ varies.5 marks