Locus of complex number satisfying $\arg \left[\right(\mathrm{z}-5+4\mathrm{i})/(\mathrm{z}+3-2\mathrm{i}\left)\right]=-\mathrm{\pi }/4$ is the arc of a circle

$\begin{array}{l}\frac{{\mathrm{z}}_0-(-3+2\mathrm{i})}{{\mathrm{z}}_0-(5-4\mathrm{i})}=\frac{\mathrm{BD}}{\mathrm{AD}}{\mathrm{e}}^{\mathrm{i\pi }/2}=\mathrm{i}\\ \Rightarrow {\mathrm{z}}_0+3-2\mathrm{i}={\mathrm{iz}}_0-5\mathrm{i}-4\\ \Rightarrow {\mathrm{z}}_0=-2-5\mathrm{i}\end{array}$

$\begin{array}{c}\Rightarrow \text{ Radius }\mathrm{AD}=|5-4\mathrm{i}-(-2-5\mathrm{i}\left)\right|\\ =|7+\mathrm{i}|\\ =\sqrt{50}=5\sqrt{2}\end{array}$

$\begin{array}{c}\text{ Length of arc }=\frac{3}{4}\text{ (Perimeter of circle) }\\ =\frac{3}{4}(2\mathrm{\pi }\times 5\sqrt{2})\\ =\frac{15\mathrm{\pi }}{\sqrt{2}}\end{array}$