Let `S={z in C:z(iz_(1)+1,|z_(1)| lt 1}`. Then, for all `z in S`, which one of the following is always true?

Options

A. `"Re"(z)-"Im"(z) lt 0`
B. `Re(z) + Im (z) `lt 0`
C. `Re(z) `lt 0`
D. Re(z)-Im(z) `gt 0`

Answer

A

Explanation

`z(iz_(1)-1)=z_(1)+1`
`rArr izz_(1)-z=z_(1)+1`
`rArr z_(1)(iz-1)=z+1`
`rArr z_(1)=(z+1)/(iz-1)`
Now, `|z_(1)| lt 1`
`|(z+1)/(iz-1)| lt 1`
`rArr |z+1| lt |z+i|`
`rArr |(x+1)+iy| lt |x+i(y+1)|`, where `z=x+iy`
`rArr (x+1)^(2)+y^(2) lt x^(2)+(y+1)^(2)`
`rArr x lt y rArr x-y lt 0 rArr "Re"(z)-"Im"(z) lt 0`.