If `z_1 and z_2` are two non zero` complex number such that`|z_1+z_2|=|z_1|+|z_2|` then `arg z_1-argz_2` is equal to (A) `- pi/2` (B) `0` (C) `-pi` (D) `pi/2`

Options

A. `-pi`
B. `-(pi)/(2)`
C. `(pi)/(2)`
D. 0

Explanation

Let `z_(1)=r_(1)(cos theta_(1)+i sin theta_(1)),z_(2)(cos theta_(2)+i sin theta_(2))`
`:. |z_(1)+z_(2)|=[(r_(1)cos theta_(1)+r_(2)cos theta_(2))+(r_(2)sin theta_(1)+r_(2) sin theta_(2))^(2)]^(1//2)`
`[r_(1)^(2)+r_(2)^(2)+2r_(1)r_(2)cos (theta_(1)-theta_(2))]^(1//2)=[(r_(1)+r_(2))^(2)]^(1//2) " " (.:'|z_(1)+z_(2)|=|z_(1)|+|z_(2)|)`
Therefore `cos (theta_(1)-theta_(2))=1rArrtheta_(1)-theta_(2)=0rArrtheta_(1)=theta_(2)`
Thus `arg(z_(1))-arg(z_(2))=0`
Trick : `|z_(1)+z_(2)|=|z_(1)|+|z_(2)|rArr z_(1), z_(2)` lies on same straight line.
`:. arg z_(1)=arg z_(2) rArr arg z_(1)=arg z_(2)=0`