Question

For a complex number $z,$ if $|z-1+i|+|z+i|=1,$ then the range of the principle argument of $z$ is
$($ Here, principle argument $\in (-\pi ,\pi ])$

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

Answer 1

The correct option is C $[\frac{-\pi }{2},\frac{-\pi }{4}]$

$|z-1+i|+|z+i|=1$
or, $|z-(1-i)|+|z-(-i)|=|(1-i)-(-i)|$
$\Rightarrow$ Locus of $z$ is the line segment joining $-i$ and $1-i$

From the above figure, minimum value of $\arg \left(z\right)=\frac{-\pi }{2}$
and Maximum value of $\text{arg}\left(z\right)=\frac{-\pi }{4}$
$\therefore$ Principle $\arg \left(z\right)\in [\frac{-\pi }{2},\frac{-\pi }{4}]$