Tangent to curve at given point

11 \includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-4_885_967_255_589} The diagram shows the curve \(y = ( 6 x + 2 ) ^ { \frac { 1 } { 3 } }\) and the point \(A ( 1,2 )\) which lies on the curve. The tangent to the curve at \(A\) cuts the \(y\)-axis at \(B\) and the normal to the curve at \(A\) cuts the \(x\)-axis at \(C\).

1. Find the equation of the tangent \(A B\) and the equation of the normal \(A C\).
2. Find the distance \(B C\).
3. Find the coordinates of the point of intersection, \(E\), of \(O A\) and \(B C\), and determine whether \(E\) is the mid-point of \(O A\).

5. Points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\), where \(p ^ { 2 } \neq q ^ { 2 }\), lie on the parabola \(y ^ { 2 } = 4 a x\).

1. Show that the chord \(P Q\) has equation $$y ( p + q ) = 2 x + 2 a p q$$ Given that this chord passes through the focus of the parabola,
2. show that \(p q = - 1\)
3. Using calculus find the gradient of the tangent to the parabola at \(P\).
4. Show that the tangent to the parabola at \(P\) and the tangent to the parabola at \(Q\) are perpendicular.