If `|z|= "max"{|z-2|,|z+2|}`, then

Options

A. `|z+barz|=2`
B. `z+barz=4`
C. `|z+barz|=1`
D. none of these

Answer

A

Explanation

We have,
`|z|="max"{|z-2|,|z+2|}`
`therefore |z|=|z-2|` or, `|z|=|z-2|`
Case I: When `|z|=|z-2|`, In this case, We have
`rArr |z|=|z-2|`
`rArr zbarz=(z-2)(barz+2)`
`rArr zbarz=(z-2)(barz-2)`
`rArr bar(zz) = bar(zz)-2z-2barz+4 rArr z+barz=2 rArr |z+barz|=2`
Case II When `|z|=|z+2|`: In this case, we have
`|z|=|z+2|`
`|z|^(2)=|z+2|^(2)`
`rArr zbarz=(z+2)(barz+2)`
`rArr zbarz=zbarz+2z+2barz+4`
`rArr z+barz=2 rArr |z+barz|=2`
Hence, in either case, we obtain
`z+barz=2` and `|z+barz|=2`