7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant.
The point \(\left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
- Find the value of \(a\).
- Show that the equation for the tangent to \(C\) at the point \(\left( 4 t ^ { 2 } , 8 t \right)\) is
$$y t = x + 4 t ^ { 2 } .$$
The tangent to \(C\) at the point \(A\) meets the tangent to \(C\) at the point \(B\) on the directrix of \(C\) when \(y = 15\).
- Find the coordinates of \(A\) and the coordinates of \(B\).