- A parabola \(C\) has equation \(y ^ { 2 } = 4 a x\) where \(a\) is a positive constant.
The point \(S\) is the focus of \(C\)
The line \(l _ { 1 }\) with equation \(y = k\) where \(k\) is a positive constant, intersects \(C\) at the point \(P\)
- Show that
$$P S = \frac { k ^ { 2 } + 4 a ^ { 2 } } { 4 a }$$
The line \(l _ { 2 }\) passes through \(P\) and intersects the directrix of \(C\) on the \(x\)-axis.
The line \(l _ { 2 }\) intersects the \(y\)-axis at the point \(A\) - Show that the \(y\) coordinate of \(A\) is \(\frac { 4 a ^ { 2 } k } { k ^ { 2 } + 4 a ^ { 2 } }\)
The line \(l _ { 1 }\) intersects the directrix of \(C\) at the point \(B\)
Given that the areas of triangles \(B P A\) and \(O S P\), where \(O\) is the origin, satisfy the ratio
$$\text { area } B P A \text { : area } O S P = 4 k ^ { 2 } : 1$$ - determine the exact value of \(a\)