N2012 P1 Q5

Referred to the origin \(O\), the points \(A\) and \(B\) have position vectors \(\mathbf a\) and \(\mathbf b\) such that \(\mathbf a=\mathbf i-\mathbf j+\mathbf k\) and \(\mathbf b=\mathbf i+2\mathbf j\).

The point \(C\) has position vector \(\mathbf c\) given by \(\mathbf c=\lambda\mathbf a+\mu\mathbf b\), where \(\lambda\) and \(\mu\) are positive constants.

1. Given that the area of triangle \(OAC\) is \(\sqrt{126}\), find \(\mu\).[4]
2. Given instead that \(\mu=4\) and that \(OC=5\sqrt3\), find the possible coordinates of \(C\).[4]