(i)[3]
Show, without using a calculator, that $w = 2 + 4i$.
(ii)[3]
It is stated that $p$ is real and $\frac{\pi}{4} \leq \arg(w + p) \leq \frac{3\pi}{4}$. Find the set of possible values of $p$.
(iii)[3]
The complex conjugate of $w$ is written as $w^*$. On an Argand diagram, $w$ and $w^*$ are shown by the points $S$ and $T$ respectively. Find the equation of the circle, in the form $|z - a| = k$, that passes through $S$, $T$ and the origin.