For any two complex numbers `z_1` and `z_2`, we have `|z_1+z_2|^2=|z_1|^2+|z_2|^2`, then

Options

A. Re`(z_(1)/z_(2))=0`
B. `"Im"(z_(1)/z_(2))=0`
C. `"Re"(z_(1)z_(2))=0`
D. `"Im"(z_(1)z_(2))=0`

Answer

A

Explanation

We have,
`|z_(1)+z_(2)|=|z_(1)|+|z_(2)|`
`rArr |z+(1)+z_(2)|^(2)=|z_(1)|^(2)+|z_(2)|^(2)`
`rArr |z_(1)|^(2) + |z_(2)|^(2)+2|z_(1)||z_(2)|cos(theta_(1)-theta_(2))=|z_(1)|^(2)+|z_(2)|^(2)`,
`rArr cos(theta_(1)-theta_(2))=0`
`rArr theta_(1)-theta_(2)=pi/2 rArr "arg"(z_(1)/z_(2))=pi/2 rArr "Re"(z_(1)/z_(2))=0`.