5. Points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\), where \(p ^ { 2 } \neq q ^ { 2 }\), lie on the parabola \(y ^ { 2 } = 4 a x\).
- Show that the chord \(P Q\) has equation
$$y ( p + q ) = 2 x + 2 a p q$$
Given that this chord passes through the focus of the parabola,
- show that \(p q = - 1\)
- Using calculus find the gradient of the tangent to the parabola at \(P\).
- Show that the tangent to the parabola at \(P\) and the tangent to the parabola at \(Q\) are perpendicular.