Question

Let $S=\{z=x+iy:|z-1+i|\ge |z|,|z|<2$, $|z+i|=|z-1|\}$. Then the set of all values of $x$, for which $w=2x+$ 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)

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


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