If arg $\left(\frac{\mathrm{z}-2\mathrm{i}}{\mathrm{z}+2\mathrm{i}}\right)=\frac{\mathrm{\pi }}{6}$ then which of the following options gives the correct locus of z?

$$\begin{gathered} \arg \left(\frac{z-2i}{z+2i}\right)=\frac{\pi }{6} \\ \frac{z-2i}{z+2i}=\frac{x+iy-2i}{x+iy+2i}=\frac{x+i\left(y-2\right)}{x+i\left(y+2\right)}\frac{x-i\left(y+2\right)}{x-i\left(y+2\right)} \\ =\frac{x^2+y^2+4+i\left(xy-2x-xy-2x\right)}{x^2+{(y+2)}^2} \\ =\frac{{\displaystyle x^2+y^2+4-4xi}}{x^2+{(y+2)}^2} \\ \arg \left(\frac{{\displaystyle z-2i}}{{\displaystyle z+2i}}\right)=\frac{{\displaystyle \pi }}{{\displaystyle 6}} \\ -\frac{4x}{x^2+y^2+4}=\frac{1}{\sqrt{3}}\Rightarrow x^2+y^2+4\sqrt{3}x+4=0 \end{gathered}$$

$\arg \left(\frac{z-2i}{z+2i}\right)=\frac{\pi }{6}(acute angle)represents major arc of a circle$