6 Prove that \(\sqrt { 2 } \cos \left( 2 \theta + 45 ^ { \circ } \right) \equiv \cos ^ { 2 } \theta - 2 \sin \theta \cos \theta - \sin ^ { 2 } \theta\), where \(\theta\) is measured in degrees.
\(7 \quad A\) and \(B\) are fixed points in the \(x - y\) plane. The position vectors of \(A\) and \(B\) are \(\mathbf { a }\) and \(\mathbf { b }\) respectively. State, with reference to points \(A\) and \(B\), the geometrical significance of
- the quantity \(| \mathbf { a } - \mathbf { b } |\),
- the vector \(\frac { 1 } { 2 } ( \mathbf { a } + \mathbf { b } )\).
The circle \(P\) is the set of points with position vector \(\mathbf { p }\) in the \(x - y\) plane which satisfy \(\left| \mathbf { p } - \frac { 1 } { 2 } ( \mathbf { a } + \mathbf { b } ) \right| = \frac { 1 } { 2 } | \mathbf { a } - \mathbf { b } |\).
- State, in terms of \(\mathbf { a }\) and \(\mathbf { b }\),
- the position vector of the centre of \(P\),
- the radius of \(P\).
It is now given that \(\mathbf { a } = \binom { 2 } { - 1 } , \mathbf { b } = \binom { 4 } { 5 }\) and \(\mathbf { p } = \binom { x } { y }\).
- Find a cartesian equation of \(P\).