7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant.
The point \(P \left( a t ^ { 2 } , 2 a t \right)\) is a general point on \(C\).
- Show that the equation of the tangent to \(C\) at \(P \left( a t ^ { 2 } , 2 a t \right)\) is
$$t y = x + a t ^ { 2 }$$
The tangent to \(C\) at \(P\) meets the \(y\)-axis at a point \(Q\).
- Find the coordinates of \(Q\).
Given that the point \(S\) is the focus of \(C\),
- show that \(P Q\) is perpendicular to \(S Q\).