Question

For two complex numbers $z_1,z_2$ the realtion $|z_1+z_2|=|z_1|+|z_2|$ hold, if

• A. $arg\left(z_1\right)=arg\left(z_2\right)$
• B. $arg\left(z_1\right)+arg\left(z_2\right)=\frac{\pi }{2}$
• C. $z_1{z}_2=1$
• D. $|z_1|=|z_2|$

Answer 1

Answer: (A)

$arg\left(z_1\right)=arg\left(z_2\right)$

Since, $|z_1+z_2|=|z_1|+|z_2|$

By squaring both sides, we get

$|z_1{|}^2+|z_2{|}^2+2|z_1||z_2|\cos ({\theta }_1-{\theta }_2)$

$=|z_1{|}^2+|z_2{|}^2+2|z_1||z_2|$

$\Rightarrow \cos ({\theta }_1-{\theta }_2)=1=\cos 0$

$\Rightarrow {\theta }_1-{\theta }_2=0$

$\Rightarrow {\theta }_1={\theta }_2$

$\Rightarrow arg\left(z_1\right)=arg\left(z_2\right)$