complex number - modulus problem

Question

If $|z_1+z_2|=|z_1|+|z_2|$ where $z_1 ; z_2$ are different non zero complex numbers, then

(a) $Re(\frac{z_1}{z_2})=0$ (b) $Im(\frac{z_1}{z_2})=0$ (c) $z_1+z_2=0$

Please guide how to proceed...

Answer 1

Square both sides to get $|z_1|^2+|z_2|^2 + z_1 \overline{z_2}+ z_2 \overline{z_1} = |z_1|^2+|z_2|^2 + |z_1||z_2|$, from which we get $\operatorname{re} z_1 \overline{z_2} = |z_1||z_2|$.

Note that if $\operatorname{re} z = |z|$, then $z = |z|$. Hence $z_1 \overline{z_2} = |z_1||z_2|$. Multiply both sides by $\frac{z_2}{|z_1||z_2|^2}$ to get $\frac{z_1}{|z_1|} = \frac{z_2}{|z_2|}$.

Since $\frac{z_1}{z_2} = \frac{|z_1|}{|z_2|}$, it follows that (b) is correct.

Answer 2

There are (at least) two strategies you can employ. One is to blindly look for $z_1,z_2$ such that $|z_1+z_2|=|z_1|+|z_2|$ and see which of the options can be eliminated. Trying you luck with very simple example, guess that $z_1=1$ and $z_2=2$. Then indeed $|z_1+z_2|=|z_1|+|z_2|$ holds, but $Re(\frac{z_1}{z_2}\ne 0$ and $z_1+z_2\ne 0$. So, if you truly believe on the options must hold, then its option 2.

The second strategy is to think about the geometric meaning of the given expressions. The equality $|z_1+z_2|=|z_1|+|z_2|$, thinking of the triangle inequality, can only hold if $z_1$ and $z_2$ are on the same line through the origin. That means that there is some real $\lambda$ with $z_1=\lambda \cdot z_2$. Then $\frac{z_1}{z_2}=\lambda$, so option two is the right answer.

Answer 3

If $|z_1+z_2|=|z_1|+|z_2|$ then $z_1$ and $z_2 $ are in the same direction(if you think of them as vectors). $\implies \Im(\dfrac{z_1}{z_2})=0$ Note that it can also be true that $z_2=0$