Question

The shaded region in Fig. 2.55 is given by

• A. $\left\{z:|z-1|<2,|\arg (z+1)|<\frac{\pi}{2}\right\}$
• B. $\left\{z:|z+1|<2,|\arg (z+1)|<\frac{\pi}{2}\right\}$
• C. $\left\{z:|z-1|>2,|\arg (z-1)|<\frac{\pi}{4}\right\}$
• D. $\left\{z:|z+1|>2,|\arg (z+1)|<\frac{\pi}{4}\right\}$

Answer 1

Answer: (D)

We have $A P=A B=A Q=2$
Thus, for the shaded region $|z+1|>2$
Also, $\angle B A Q=\tan ^{-1}\left(\frac{\sqrt{2}}{\sqrt{2}-1+1}\right)=\frac{\pi}{4}$ and $\angle B A P=-\pi / 4$
Hence, for the shaded region $|z+1|>2$ and $\operatorname{larg}(z+1) \mid<\pi / 4$