If `|z-1|=1,` where z is a point on the argand plane, show that`(z-2)/(z)=i tan (argz),where i=sqrt(-1).`

Options

A. tan(arg)z
B. cot(arg z)
C. itan (arg z)
D. none of these

Answer

C

Explanation

We have, `|z-1|=1`
So, let `z-1=costheta+isintheta`
`rArr z-2=sintheta//2(-sintheta//2+icostheta/2)`
and `z=2costheta//2(costheta//2+isintheta//2)`
and, `z=2costheta//2(costheta//2+isintheta//2)`
`rArr (z-2)/(z)=i{(costheta/2+isintheta/2)/(costheta/2+isintheta/2)}tantheta/2`
`rArr (z-2)/z = itantheta/2 rArr (z-2)/z=i tan(arg z)`