Question

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`

Answer 1

Correct Answer - 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)`