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If a complex number z satisfies |z|^(2)+...

If a complex number `z` satisfies `|z|^(2)+(4)/(|z|)^(2)-2((z)/(barz)+(barz)/(z))-16=0`, then the maximum value of `|z|` is

A

`sqrt(6)+1`

B

`4`

C

`2+sqrt(6)`

D

`6`

Text Solution

Verified by Experts

The correct Answer is:
C
`(c )` Let `z=r(costheta+isintheta)`
Then the given equation is `r^(2)+(4)/(r^(2))-2.2cos2theta-16=0`
`implies r^(4)-4r^(2)(cos2theta+4)+4=0`
`implies r^(2)=2(cos2theta+4)+2sqrt((cos2theta+4)^(2)-1)`
The maximum value is obtained when `cos2theta=1`
`:.` The maximum value of `r^(2)=10+2sqrt(24)`
`=(2+sqrt(6))^(2)`
`implies` The maximum value of `r=2+sqrt(6)`
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