2022 CJC P2 Q4

Referred to the origin \(O\), the fixed points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively, where \(\mathbf{a}\) and \(\mathbf{b}\) are non-zero vectors. It is given that \(O\) divides the line segment \(AC\) in the ratio \(1:3\). Point \(D\) divides the line segment \(AB\) in the ratio \(\lambda :1-\lambda \). Point \(E\) divides the line segment \(BC\) in the ratio \(\mu :1-\mu \).

Show that the area of triangle \(ODE\) can be written as \(k\left| \mathbf{a}\times \mathbf{b} \right|\), where \(k\) is a constant to be found in terms of \(\mu \) and \(\lambda \).[5]

A variable point \(S\) has position vector \(\mathbf{s}\) relative to the origin \(O\). Given that \(\mathbf{s}\times \mathbf{b}=-\mathbf{b}\times \mathbf{a}\), describe geometrically the set of all possible positions of the point \(S\).[3]