- The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\).
The straight line \(l\) passes through the point \(S\) and meets the directrix of \(C\) at the point \(D\).
Given that the \(y\) coordinate of \(D\) is \(\frac { 24 a } { 5 }\),
- show that an equation of the line \(l\) is
$$12 x + 5 y = 12 a$$
The point \(P \left( a k ^ { 2 } , 2 a k \right)\), where \(k\) is a positive constant, lies on the parabola \(C\).
Given that the line segment \(S P\) is perpendicular to \(l\), - find, in terms of \(a\), the coordinates of the point \(P\).