If `|2z-1|=|z-2|a n dz_1, z_2, z_3` are complex numbers such that `|z_1- (alpha)|< alpha,|z_2-beta||z|` d. `>2|z|`

Options

A. `lt|z|`
B. `lt2|z|`
C. `gt|z|`
D. ` gt2|z|`

Answer

B

Explanation

`|2z-1\|=|z-2|`
or `|2z-1|^(2)=|z-2|^(2)`
`or (2z-1)(2barz-1)=(z-2)(barz-2)`
`or 4zbarz-2barz-2z+1=zbarz-2barz-2z+4`
`or 3|z|^(2)=3`
`or |z|=1`
Again
`|z_(1)+z_(2)|=|z_(1)-alpha+z_(2)-beta+alpha+beta|`
`le|z_(1)-alpha|+|z_(2)-beta|+|alpha+beta|`
`ltalpha+beta+|alpha+beta|`
`=2|alpha+beta|" " [therefore alpha,betagt0]`
`therefore |(z_(1)+z_(2))/(alpha+beta)|lt2`
`or |(z_(1)+z_(2))/(alpha+beta)|lt2|z|`