- A circle \(C\) in the complex plane has equation
$$| z - ( - 3 + 3 i ) | = \alpha | z - ( 1 + 3 i ) |$$
where \(\alpha\) is a real constant with \(\alpha > 1\)
Given that the imaginary axis is a tangent to \(C\)
- sketch, on an Argand diagram, the circle \(C\)
- explain why the value of \(\alpha\) is 3
The circle \(C\) is contained in the region
$$R = \left\{ z \in \mathbb { C } : \beta \leqslant \arg z \leqslant \frac { \pi } { 2 } \right\}$$
- Determine the maximum value of \(\beta\)
Give your answer in radians to 3 significant figures.