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If z(1) = 5 + 12i and |z(2)| = 4, then...

If `z_(1) = 5 + 12i` and `|z_(2)| = 4`, then

A

maximum `(|z_(1)+ iz_(2)|) = 17`

B

minimum `(|z_(1)+(1+i)z_(2)|) = 13 - 4sqrt(2)`

C

minimum`|(z_(1))/(z_(2)+(4)/(z_(2)))|=(13)/(4)`

D

maximum `|(z_(1))/(z_(2) + (4)/(z_(2)))|=(13)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, D
`z_(1) =5 + 12i, |z_(2)| = 4`
`|z_(1) + iz_(2)|le |z_(1)|= 13 + 4 =17`
`therefore|z_(1)+(1+i)z_(2)|le||z_(1)|-|1+i||z_(2)||= 13 - 4sqrt(2)`
`therefore min (|z_(1)+(1+i)z_(2)|) = 13 - 4sqrt(2)`
`|z_(2)+(4)/(z_(2))| le|z_(2)|+(4)/(|z_(2)|)=4 +1=5`
`|z_(2)+(4)/(z_(2))| ge|z_(2)| +(4)/(|z_(2)|)=4 - 1=5`
`therefore max|(z_(1))/(z_(2)+(4)/(z_(2)))|= (13)/(3) and min |(z_(1))/(z_(2)+(4)/(z_(2)))| = (13)/(5)`
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