If `|z-1| =1` and arg`(z)=theta`, where `z ne 0` and `theta` is acute, then `(1-2/z)` is equal to

Options

A. `tantheta`
B. `I tantheta`
C. `tan theta/2`
D. `I tantheta/2`

Answer

B

Explanation

We have,
`|z-1|=1`
`rArr |z-1(1+0i)|=1`
`rArr` z lies on the circle with center at (1,0) and radius 1.

It is given that `theta` is acute and P(z) is a point on the circle `|z-1|=1` such that arg (z) `=theta`, i.e. `angleAOP=theta`
Now, `anlgeOPA=pi/2`
`rArr (2-z)=(PA)/(PO)(0-z)e^(ipi//2)`
`rArr (z-2)/z=i tantheta [therefore tantheta=(PA)/PO)]`
`rArr (z-2)/z=itantheta`