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 $\left(Z\right)\in \left[-\frac{\pi }{2},-\frac{\pi }{4}\right]$