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

Options

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

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 center `z_0` and radius r is `|z-z_0|=r`
`therefore |z-(-1 + 0i ) | gt 2`
`rArr | z + 1| gt 2 `
Also ,argument of z+1 with respect to positive direction of X-axis is `pi//4`
`therefore arg( z+1) le (pi)/4`
and argument of z+1 is anticlock wise dierction is `-pi//4`
`therefore - pi//4 le arg (z+1)`
From Eqs (i) and (ii)
`|arg (z+1)| le pi//4`