- The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a non-zero constant.
The point \(P \left( c p , \frac { c } { p } \right)\), where \(p \neq 0\), lies on \(H\).
- Use calculus to show that an equation of the normal to \(H\) at \(P\) is
$$p ^ { 3 } x - p y + c \left( 1 - p ^ { 4 } \right) = 0$$
The normal to \(H\) at the point \(P\) meets \(H\) again at the point \(Q\).
- Find the coordinates of the midpoint of \(P Q\) in terms of \(c\) and \(p\), simplifying your answer where possible.