If `|omega|=2`, then the set of points `z=omega-1/omega` is contained in or equal to the set of points z satisfying

Options

A. `"Im"(z)=0`
B. `|"Im"(z)| le 1`
C. `|"Re"(z)| le 2`
D. `|z| le 3`

Answer

D

Explanation

We have,
`|omega|=2 rArr omega=2(costheta+isintheta)`
`therefore z=omega-1/omega=2(costheta+sintheta)-1/2(costheta-isintheta)`
`rArr z=3/2costheta+5/2isintheta`
`rArr |z|=sqrt(9/4+25/4)=sqrt(17)/2 lt 3`
So, option (d) is correct.
Again, `z=3/2costheta+5/2isintheta`
`rArr x+iy=3/2costheta+5/2isintheta,` where `z=x+iy`
`rArr x=3/2costheta,y=5/2sintheta`
`rArr (2x)/3=costheta,(2y)/5=sintheta`
`=4/9x^(2)+4/25y^(2)=1`
`rArr 25x^(2)+9y^(2)=225/4`
`rArr` z lies on the ellipse `25x^(2)+9y^(2)=225/4` whose eccentricity e is given by `e=sqrt(1-9/25)=4/5`