Let $\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|

Question

Let $\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|<2,|z+i|=|z-1|\}$. Then the set of all values of $x$, for which $w=2 x+i y \in \mathrm{S}$ for some $y \in \mathbb{R}$, is :

$\left(-\sqrt{2}, \frac{1}{2 \sqrt{2}}\right]$

$\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right]$

$\left(-\sqrt{2}, \frac{1}{2}\right]$

$\left(-\frac{1}{\sqrt{2}}, \frac{1}{2 \sqrt{2}}\right]$

Solution

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