Edexcel F1 2016 January — Question 6

Exam BoardEdexcel
ModuleF1 (Further Pure Mathematics 1)
Year2016
SessionJanuary
TopicConic sections

6. 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 \(H\).
  1. Show that the normal to \(H\) at \(P\) has equation $$y p - p ^ { 3 } x = c \left( 1 - p ^ { 4 } \right)$$ The normal to \(H\) at \(P\) meets \(H\) again at the point \(Q\).
  2. Find, in terms of \(c\) and \(p\), the coordinates of \(Q\).