Question

If $\left|z-1+i\right|+\left|z+i\right|=1$ then find range of principle argument of z.

Answer 1

$|z-1+i|+|z+i|=1$

$z-1+i+z+i=1$

$2z+2i=2$

$z+i=1$

$z=1-i$

$arg\left(z\right)\theta ={\tan }^{-1}\frac{b}{a}={\tan }^{-1}\frac{-1}{1}={\tan }^{-1}-1=-{45}^{\circ }$

$\theta ={180}^{\circ }-{45}^{\circ }={135}^{\circ }$

$Rangeis[-{45}^{\circ },{135}^{\circ }]$