2020 HCI Promo Q9

With respect to an origin \(O\), the points \(A\),\(B\),\(C\) and \(D\) have position vectors \(\mathbf{a}=4\mathbf{i}\), \(\mathbf{b}=2\mathbf{k}\), \(\mathbf{c}=3\mathbf{i}-5\mathbf{j}\) and \(\mathbf{d}=3\mathbf{i}+11\mathbf{j}+\left( \alpha +2 \right)\mathbf{k}\) where \(\alpha \) is a constant, respectively.

1. The point \(P\) on \(OC\) is such that \(AP\) gives the shortest distance from \(A\) to the line \(OC\). Find the exact position vector of \(P\). Hence state the ratio of the area of triangle \(OAP\) to the area of triangle \(OAC\) in the form of \(x:y\) where \(x\), \(y\in {{\mathbb{Z}}^{+}}\).[3]
2. The point \(M\) is the midpoint of \(AB\) and the point \(E\) on \(CM\) is such that \(ME:EC=\lambda :1-\lambda \). Given that the point \(N\) lies on \(BE\) produced such that \(A\),\(N\) and \(C\) are collinear and \(AN=2NC\), find the value of \(\lambda \).[6]
3. Given that the line passing through \(A\) and \(C\) and the line passing through \(B\) and \(D\) are skew lines, what can be said about the value of \(\alpha \)?[3]