4. The rectangular hyperbola \(H\) has Cartesian equation \(x y = 4\)
The point \(P \left( 2 t , \frac { 2 } { t } \right)\) lies on \(H\), where \(t \neq 0\)
- Show that an equation of the normal to \(H\) at the point \(P\) is
$$t y - t ^ { 3 } x = 2 - 2 t ^ { 4 }$$
The normal to \(H\) at the point where \(t = - \frac { 1 } { 2 }\) meets \(H\) again at the point \(Q\).
- Find the coordinates of the point \(Q\).