Problem

Given that

$\frac{3z - 1}{2} = \frac{\lambda + 5i}{\lambda - 4i}$

where $\lambda$ is a real constant,

(a) determine $z$, giving your answer in the form $x + yi$, where $x$ and $y$ are real and in terms of $\lambda$.

Given also that $\arg z = \dfrac{\pi}{4}$

(b) find the possible values of $\lambda$.