I proceeded by plotting $z$ on the complex plane, and the modulus of $z-3-4i$:
From this I deduced that: $$\begin{align}&\mathrm{Re}(z)=5\cos\theta+3\\&\mathrm{Im}(z)=(5\sin\theta+4)i\end{align}$$
And hence that: $$\begin{align}z&=\mathrm{Re}(z)+\mathrm{Im}(z)\\&=(5\cos\theta+3)+(5\sin\theta+4)i\end{align}$$
Which may be incorrect.
Regardless, I cannot derive from it the equation of the subset.