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 $AP=AB=AQ=2$
Thus, for the shaded region $|z+1|>2$
Also, $\angle BAQ={\tan }^{-1}\left(\frac{\sqrt{2}}{\sqrt{2}-1+1}\right)=\frac{\pi }{4}$ and $\angle BAP=-\pi /4$
Hence, for the shaded region $|z+1|>2$ and $\mathrm{larg}(z+1)\mid <\pi /4$