$$\begin{array}{lrl} \text { Given that, } & & z=(1+i \sqrt{3})^2 \\ \Rightarrow & & z=1-3+2 i \sqrt{3} \Rightarrow z=-2+i 2 \sqrt{3} \\ \Rightarrow & & \tan \alpha=\left|\frac{2 \sqrt{3}}{-2}\right|=|-\sqrt{3}|=\sqrt{3} \left[\because \tan \alpha=\left|\frac{\operatorname{lm}(z)}{\operatorname{Re}(z)}\right|\right]\\ \Rightarrow & & \tan \alpha=\tan \frac{\pi}{3} \Rightarrow \alpha=\frac{\pi}{3} \\ \because & & \operatorname{Re}(z)<0 \text { and } \operatorname{lm}(z)>0 \\ \Rightarrow & & \arg (z)=\pi-\frac{\pi}{3} \Rightarrow=\frac{2 \pi}{3} \end{array}$$

Let $$z=x+i y$$

Given that, $\quad\left|\frac{z-5 i}{z+5 i}\right|=\left|\frac{x+i y-5 i}{x+i y+5 i}\right|$

$$\begin{array}{ll} \Rightarrow & \left|\frac{z-5 i}{z+5 i}\right|=\frac{|x+i(y-5)|}{|x+i(y+5)|} \\ \Rightarrow & \left|\frac{z-5 i}{z+5 i}\right|=\frac{\sqrt{x^2+(y-5)^2}}{\sqrt{x^2+(y+5)^2}} \quad \left[\because\left|\frac{z-5 i}{z+5 i}\right|=1\right] \end{array}$$

On squaring both sides, we get

$$\begin{array}{rlrl} & x^2+(y-5)^2 =x^2+(y+5)^2 \\ \Rightarrow & -10 y =+10 y \\ \Rightarrow & 20 y =0 \\ \therefore & y =0 \end{array}$$

So, $z$ lies on real axis.