- The rectangular hyperbola \(H\) has equation \(x y = 64\)
The point \(P \left( 8 p , \frac { 8 } { p } \right)\), where \(p \neq 0\), lies on \(H\).
- Use calculus to show that the normal to \(H\) at \(P\) has equation
$$p ^ { 3 } x - p y = 8 \left( p ^ { 4 } - 1 \right)$$
The normal to \(H\) at \(P\) meets \(H\) again at the point \(Q\).
- Determine, in terms of \(p\), the coordinates of \(Q\), giving your answers in simplest form.
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