If the point $P$ denotes the complex number $z=x+i y$ in the argand plane and $\frac{z-(2-i)}{z+(1+2 i)}$ is purely imaginary number, then the locus of $P$ is

If $(\sqrt{3}-i)^n=2^n, n \in N$, then the least possible value of $n$ is

$$ (1+\sqrt{5}+i \sqrt{10-2 \sqrt{5}})^5= $$