2025 EJC P2 Q1

The curve \(C\) has equation \(x^3 + y^3 - 3xy = k\), where \(k\) is a positive constant.
\(C\) intersects the \(x\)-axis at the point \(A\).

1. Find the equation of the tangent to \(C\) at \(A\). Give your answer in the form \(y = px + q\), where \(p\) and \(q\) are constants to be found in terms of \(k\).[5]
2. There are two points on \(C\) where the tangents are parallel to the \(y\)-axis. Given that \(k = 3\), find the exact \(y\)-coordinates of these two points.[3]