2019 EJC Promo Q4

Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively.

1. Given that non-zero numbers \(\lambda \) and \(\mu\) are such that \(\lambda \mathbf{a}+\mu\mathbf{b}+\mathbf{c}=\mathbf{0}\) and \(\lambda +\mu +1=0\) with \(\mu >0\). Show that \(A\), \(B\), and \(C\) are collinear and find the ratio \(CA:CB\) in terms of \(\mu \).[4]
2. \(F\) is another point such that the line passing through \(A\), \(B\) and \(C\) does not contain it. Find \(\frac{\left| \overrightarrow{BF}\times \overrightarrow{BC} \right|}{\left| \overrightarrow{AF}\times \overrightarrow{AC} \right|}\) in terms of \(\mu \).[2]