NJC IP4 Tangents and Normals Tutorial Q2

The equation of a curve is \(y=x^3\). The distinct points \(P\) and \(Q\) lying on the curve have \(x\)-coordinates \(p\) and \(q\) respectively, where \(p\) and \(q\) are real constants.

1. Given that the tangent to the curve at \(Q\) passes through point \(P\), show that\[p^2 + pq + cq^2 = 0\] for some constant \(c\) to be determined. Hence, find \(p\) in terms of \(q\).[6]
2. Justify whether there exists a point on the curve such that the normal to the curve at this point cuts the curve again at another point.[4]