- The parabola \(P\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant.
The point \(A \left( a t ^ { 2 } , 2 a t \right)\), where \(t \neq 0\), lies on \(P\).
- Use calculus to show that an equation of the tangent to \(P\) at \(A\) is
$$y t = x + a t ^ { 2 }$$
The point \(B \left( 2 k ^ { 2 } , 4 k \right)\) and the point \(C \left( 2 k ^ { 2 } , - 4 k \right)\), where \(k\) is a constant, lie on \(P\).
The tangent to \(P\) at \(B\) and the tangent to \(P\) at \(C\) intersect at the point \(D\).
Given that the area of the triangle \(B C D\) is 432 - determine the coordinates of \(B\) and the coordinates of \(C\).