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Prove that |z(1)|+|z(2)|=|(1)/(2)(z(1)+z...

Prove that `|z_(1)|+|z_(2)|=|(1)/(2)(z_(1)+z_(2))+sqrt(z_1z_2)|+|(1)/(2)(z_(1)+z_(2))-sqrt(z_(1)z_(1))|.`

A

`|z_(1)+z_(2)|`

B

`|z_(1)-z_(2)|`

C

`|z_(1)|+|z_(2)|`

D

`|z_(1)|-|z_(2)|`

Text Solution

Verified by Experts

The correct Answer is:
C
We have,
`|1/2(z_(1)+z_(2))+sqrt(z_(1)z_(2))|+1/2|(z_(1)+z_(2))-sqrt(z_(1)z_(2))|`
`=1/2|z_(1)+z_(2)+2sqrt(z_(1)z_(2))|+1/2|z_(1)+z_(2)-2sqrt(z_(1)z_(2))|`
`=1/2{|(sqrt(z_(1))+sqrt(z_(2)))|^(2)+|sqrt(z)-sqrt(z_(2))|^(2)}`
`=1/2{2|sqrt(z_(1))|^(2)+|sqrt(z_(2)|^(2))}=|z_(1)|+|z_(2)|`
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OBJECTIVE RD SHARMA ENGLISH-COMPLEX NUMBERS -Chapter Test
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