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 :

Options

A. `|z+1| gt 2, |"arg"(z+1)| lt pi/4`
B. `|z+1| lt 2,|"arg"(z+1)| lt pi/4`
C. `|z-1| gt 2, |"arg"(z+1)| gt pi/4`
D. `|z-1|lt 2, |"arg"(z+1)| gt pi/2`

Answer

A

Explanation

We have,
`PQ=PR=PS`
`therefore` QR is an arc of the circle centred at `P(-1,0)` and radius 2 units.
Also, `angleQPS=angleRPS=pi/4`
Therefore, any point z in the shaded region lies outside the circle `|z+1|=2` and the line joining z and `P(-1,0)` makes angle less than `45^(@)` with X-axis on its either side.
Hence, `|z+1| gt 2` and `|"arg"(z+1)| lt pi/4`.