Question

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

• A. $\left(-\sqrt{2}, \frac{1}{2 \sqrt{2}}\right]$
• B. $\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right]$
• C. $\left(-\sqrt{2}, \frac{1}{2}\right]$
• D. $\left(-\frac{1}{\sqrt{2}}, \frac{1}{2 \sqrt{2}}\right]$

Answer 1

Answer: (B)

$|z-1+i| \geq|z| ;|z|< 2 ;|z+i|=|z-1|$


Hence
$w=2 x+i y \in S $
$ 2 x \leq \frac{1}{2} \therefore x \leq \frac{1}{4}$
Now
$ (2 x)^2+(2 x)^2< 4 $
$ x^2<\frac{1}{2} \Rightarrow x \in\left(\frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$
$ \therefore x \in\left(\frac{-1}{\sqrt{2}}, \frac{1}{4}\right]$