2018 TGM P1 Q10

The parametric equations of a curve are\[x = t^2, y = t^2 - t.\]

1. The point \(P\) on the curve has parameter \(p\). Show that the equation of the tangent at \(P\) is \(2py = (2p - 1)x - p^2\).[3]
2. The tangent at \(P\) meets the \(x\)- and \(y\)- axes at the points \(Q\) and \(R\) respectively. Find the coordinates of \(Q\) and \(R\) in terms of \(p\).[2]
3. Find a cartesian equation of the locus of the midpoint of \(QR\) as \(p\) varies.[3]
4. Find the equation of the tangent at the point \((4, 6)\) and determine if this tangent meets the curve again.[4]