6. A parabola \(C\) has equation \(y ^ { 2 } = 4 a x , \quad a > 0\)
The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\) lie on \(C\), where \(p \neq 0 , q \neq 0 , p \neq q\).
- Show that an equation of the tangent to the parabola at \(P\) is
$$p y - x = a p ^ { 2 }$$
- Write down the equation of the tangent at \(Q\).
The tangent at \(P\) meets the tangent at \(Q\) at the point \(R\).
- Find, in terms of \(p\) and \(q\), the coordinates of \(R\), giving your answers in their simplest form.
Given that \(R\) lies on the directrix of \(C\),
- find the value of \(p q\).