Question

For a complex number $Z$ , if $\left|Z - 1 + i\right|+\left|Z + i\right|=1,$ then the range of the principle argument of $Z$ is (where principle $arg\left(Z\right)\in \left(- \pi , \pi \right]$ )

• A. $\left[- \frac{\pi }{4} , \frac{\pi }{4}\right]$
• B. $\left[\frac{\pi }{4} , \frac{\pi }{2}\right]$
• C. $\left[- \frac{\pi }{2} , - \frac{\pi }{4}\right]$
• D. $\left[- \frac{\pi }{2} , \frac{\pi }{2}\right]$

Answer 1

Answer: (C)

$\left|Z - 1 + i\right|+\left|Z + i\right|=\left|\left(1 - i\right) - \left(- i\right)\right|$
$\Rightarrow $ locus of $Z$ is the line segment joining $-i$ with $1-i$

$\Rightarrow $ minimum arg $\left(Z\right)=-\frac{\pi }{2}$ & maximum arg $\left(Z\right)=-\frac{\pi }{4}$
$\Rightarrow $ principal arg $\left(Z\right)\in \left[- \frac{\pi }{2} , - \frac{\pi }{4}\right]$