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If |z-(1/z)|=1, then a. (|z|)(m a x)=(...

If `|z-(1/z)|=1,` then a. `(|z|)_(m a x)=(1+sqrt(5))/2` b. `(|z|)_(m in)=(sqrt(5)-1)/2` c. `(|z|)_(m a x)=(sqrt(5)-2)/2` d. `(|z|)_(m in)=(sqrt(5)-1)/(sqrt(2))`

A

`|z|_("max") = (1+sqrt(5))/(2)`

B

`|z|_("min")=(sqrt(5)-1)/(2)`

C

`|z|_("max") = (sqrt(4)-2)/(2)`

D

`|z|_("min") = (sqrt(5)-1)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
A, B
`|z-(1)/(z)|=1`
`rArr 1ge||z|-(1)/(|z|)|`
`rArr -1le|z|-(1)/(|z|)le1`
`rArr -|z|le|z|^(2) -1 le|z|`
Form `|z|^(2) -1le|z|`, we get
`|z|^(2)+|z| -1ge0`
`rArr |z|ge (-1 + sqrt(5))/(2)`
Form`|z|^(2) - 1 le |z|`, we get
`|z|- |z|-1 le 0`
`rArr (1-sqrt(5))/(2) le |z|le (1+sqrt(5))/(2)" "(2)`
From (1) and (2), we get
`rArr (-1+sqrt(5))/(2) le|z|le(1+sqrt(5))/(2)`
`rArr |z|_(min) = (sqrt(5)-1)/(2) , |z|_(min) =(1+sqrt(5))/(2)`
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