The range of real number $\text{'}\alpha \text{'}$  for which the equation $Z-\alpha |z-1|+2i=0$  has a solution

$\begin{array}{l}z=x+iy\\ x+iy+2i=2|z-1|=\alpha \left|(x-1)+iy\right|\\ x+i(y+2)=\alpha \sqrt{{(x-1)}^2+y^2}\\ y+2=0\\ x=\alpha \sqrt{{(x-1)}^2+4}\\ x^2={\alpha }^2(x^2-2x+5)\\ x^2({\alpha }^2-1)-2{\alpha }^{2}x+5{\alpha }^2=0\\ x\in R,\Delta \ge 0\Rightarrow b^2-4ac\ge 0\Rightarrow 4{\alpha }^2(5-4{\alpha }^2)\ge 0\end{array}$