Without using a calculator, solve the equation
$$3 w + 2 \mathrm { i } w ^ { * } = 17 + 8 \mathrm { i }$$
where \(w ^ { * }\) denotes the complex conjugate of \(w\). Give your answer in the form \(a + b \mathrm { i }\).
In an Argand diagram, the loci
$$\arg ( z - 2 \mathrm { i } ) = \frac { 1 } { 6 } \pi \quad \text { and } \quad | z - 3 | = | z - 3 \mathrm { i } |$$
intersect at the point \(P\). Express the complex number represented by \(P\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), giving the exact value of \(\theta\) and the value of \(r\) correct to 3 significant figures.