2013 AJC P2 Q2

The points \(A\), \(B\) and \(C\) lie on a circle with center \(O\) and diameter \(AC\). It is given that \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\).

1. Find \(\overrightarrow{BC}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). Hence show that \(AB\) is perpendicular to \(BC\).[4]
2. Show that the area of triangle \(ABC\) can be written as \(k\left| \mathbf{a}\times \mathbf{b} \right|\) where \(k\) is a constant to be found. Hence find, in terms of \(\left| \mathbf{a} \right|\), the maximum area of triangle \(ABC\).[4]
3. Given that \(\measuredangle AOB=30{}^\circ \), find \(\overrightarrow{OF}\) where \(F\) is the foot of perpendicular of \(B\) to \(AC\). Hence, find \(\overrightarrow{OB'}\) where \(B'\) is the reflection of \(B\) along the line \(AC\).[3]