Question

If `z_1a n dz_2` are complex numbers and `u=sqrt(z_1z_2)` , then prove that `|z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u|`

Answer 1

`|(z_(1)+z_(2))/(2)+u|+|(z_(1) +z_(2))/(2) -u|`
`=|(z_(1)+z_(2))/(2) +sqrt(z_(1)z_(2))|+|(z_(1)+z_(2))/(2) =sqrt(z_(1)z_(2))|`
`=|((sqrt(z_(1))+sqrt(z_(2))^(2)))/(2)|+|((sqrt(z_(1))-sqrtz_(2)))/(z)|`
`|((p+q)^(2))/(2)|+|((p-q)^(2))/(2)|" "("Where" p = sqrt(z_(1)) and q = sqrt(z_(2)))`
`= (1)/(2) [|p+q|^(2) +|p-q|^(2)]`
`=(1)/(2) [2|p|^(2)+2|q|^(2)]`
`|p|^(2) +|q|^(2)`
`=|p^(2)| +|q^(2)|`
`=|z_(1)| +|z_(2)|`