Consider the complex numbers `z_(1)` and `z_(2)` Satisfying the relation `|z_(1)+z_(2)|^(2)=|z_(1)|^(2) + |z_(2)|^(2)` Possible difference between the argument of `z_(1)` and `z_(2)` is

Options

A. 0
B. `pi`
C. `-(pi)/(2)`
D. none of these

Answer

C

Explanation

Given that
`|z_(1) + z_(2)|^(2) = |z_(1)|^(2) + |z_(2)|^(2)`
`rArr |z_(1)|^(2) + |z_(2)|^(2) + z_(1)barz_(1) + barz_(1)z_(2) = |z_(1)|^(2)+|z_(2)|^(2)`
`rArr z_(1)barz_(1) + barz_(1)z_(2) = 0" "(1)`
`rArr (z_(1))/(z_(2)) + (barz_(1))/(barz_(2))" "("dividinbg by " z_(2)barz_(2))`
`rArr (z_(1))/(z_(2))+bar((z_(1)/(z_(2)))) = 0" "(2)`
From (1), `z_(2) barz_(2)` is purely imaginary. From (2) `z_(1)//z_(2)` is purely imaginary. Hence,
`arg((z_(1))/(z_(2))) = pm (pi)/(2) or arg(z_(1)) - arg(z_(2)) = pm(pi)/(2)`
Also, `i(z_(1)//z_(2))` is purely real. Hence its possible arguments are 0 and `pi`.