6. The parabola \(C\) has Cartesian equation \(y ^ { 2 } = 8 x\)
The point \(P \left( 2 p ^ { 2 } , 4 p \right)\) and the point \(Q \left( 2 q ^ { 2 } , 4 q \right)\), where \(p , q \neq 0 , p \neq q\), are points on \(C\).
- Show that an equation of the normal to \(C\) at \(P\) is
$$y + p x = 2 p ^ { 3 } + 4 p$$
- Write down an equation of the normal to \(C\) at \(Q\)
The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(N\)
- Show that \(N\) has coordinates
$$\left( 2 \left( p ^ { 2 } + p q + q ^ { 2 } + 2 \right) , - 2 p q ( p + q ) \right)$$
The line \(O N\), where \(O\) is the origin, is perpendicular to the line \(P Q\)
- Find the value of \(( p + q ) ^ { 2 } - 3 p q\)