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Complex numbers `z_(1)` and `z_(2)` satisfy `|z_(1)|=2` and `|z_(2)|=3`. If the included angle of their corresponding vectors is `60^(@)`, then the value of `19|(z_(1)-z_(2))/(z_(1)+z_(2))|^(2)` is

A

`5`

B

`6`

C

`7`

D

`8`

Text Solution

Verified by Experts

The correct Answer is:
C
`(c )`
`OC=|z_(1)+z_(2)|` and `AB=|z_(1)-z_(2)|`
`|z_(1)+z_(2)|=sqrt(|z_(1)|^(2)+|z_(2)|^(2)-2|z_(1)||z_(2)|cos120^(@))`
`=sqrt(4+9+2.3)=sqrt(19)`
and `|z_(1)-z_(2)|=sqrt(|z_(1)|^(2)+|z_(2)|^(2)-2|z_(1)||z_(2)|cos60^(@))`
`=sqrt(4+9-6)=sqrt(7)`
`:.|(z_(1)-z_(2))/(z_(1)+z_(2))|=sqrt((7)/(19))`
`:.19|(z_(1)-z_(2))/(z_(1)+z_(2))|^(2)=7`
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