If z is a complex number satisfying `|z^(2)+1|=4|z|`, then the minimum value of `|z|` is

Options

A. `2sqrt(5)+4`
B. `2sqrt(5)-4`
C. `sqrt(5)-2`
D. none of these

Answer

C

Explanation

We have,
`|z_(2)+|=4|z|`
`rArr |(z^(2)+1)/(z)|=4`
`rArr ||z|-|1/z|| le 4` `[therefore ||z|-|1/z|| le |z+1/z|]`
`rArr ||z|^(2)-1| le 4|z|`
`rArr -4|z| le {|z|^(2)-1} le 4|z|`
`rArr |z|^(2)-4|z|-1 le 0` or `|z|^(2)+4|z|-1 ge 0`
Case I When `|z|^(2) + 4|z|-1 ge 0`
In this case, we have
`|z|^(2)+4|z|-1 ge 0`
`rArr |z| ge -2 +sqrt(5)`
`rArr` Minimum value of `|z|` is `sqrt(5)-2`
Case II: When `|z|^(2)-4|z|-1 lt 0`
In this case, we have
`|z|^(2)-4|z|-1 le 0`
`rArr 2-sqrt(5) lt |z| lt 2+sqrt(5)`.