Let z satisfies arg (z – 3 – 4i) = arg (z – 1 + 2i) + arg (iz) + arg ($\overline{z}$ ) and z₁ is a fixed complex number such that $\left|\mathrm{z}-{\mathrm{z}}_1\right|$ is constant then

$\arg \left(\frac{\mathrm{z}-(3+4\mathrm{i})}{\mathrm{z}-(1-2\mathrm{i})}\right)=\frac{\mathrm{\pi }}{2}$
$\text{ So }{\mathrm{z}}_{\mathrm{c}}=\frac{3+4\mathrm{i}+1-2\mathrm{i}}{2}=2+\mathrm{i}={\mathrm{z}}_1$

$\left|\mathrm{z}-{\mathrm{z}}_1\right|$ is a constant

Z lies on a semi circle with centre  2+i

$$\begin{gathered} z_1=2+i\Rightarrow z_1-2=i \\ \therefore Arg(z_1-2)=\frac{\mathrm{\pi }}{2} \end{gathered}$$
$\left|z_1\right|=\left|2+i\right|\Rightarrow \left|z_1\right|=\sqrt{5}\Rightarrow \left|{\mathrm{z}}_1^8\right|=625$