- The point \(P \left( a p ^ { 2 } , 2 a p \right)\), where \(a\) is a positive constant, lies on the parabola with equation
$$y ^ { 2 } = 4 a x$$
The normal to the parabola at \(P\) meets the parabola again at the point \(Q \left( a q ^ { 2 } , 2 a q \right)\)
- Show that
$$q = \frac { - p ^ { 2 } - 2 } { p }$$
- Hence show that
$$P Q ^ { 2 } = \frac { k a ^ { 2 } } { p ^ { 4 } } \left( p ^ { 2 } + 1 \right) ^ { n }$$
where \(k\) and \(n\) are integers to be determined.