Question

The shaded region, where $P = (-1,0), Q = (-1 + \sqrt 2,\sqrt 2 )R = (-1 + \sqrt 2,-\sqrt 2), S = (1,0)$ is represented by

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

Answer 1

Answer: (A)

| z + 1| >2,| arg (z + 1) |

Answer 2

Since, | PQ | = | PS | = | PR | = 2
$\therefore \, \, \,$ Shaded part represents the external part of circle
having centre (-1,0) and radius 2.
As we know equation of circle having centre z$_0$ and
radius r, is | z - $z_0$| = r
$\Rightarrow \, \, \, \, |z+1|>2$
Also, argument of z + 1 with respect to positive direction
of X-axis is $\pi/4$
\therefore \, \, \, \, \, \, \, arg(z+1) \le \frac{\pi}{4} \hspace25mm ..(i)
and argument of z + 1 in anticlockwise direction is $-\pi /4$
\therefore \, \, \, \, \, \, \, \, -\pi/4 \le \, arg (z + 1) \hspace25mm...(ii)
From Eqs. (i) and (ii),
|arg (2 + 1) | $\le \pi/4$