Question

Let $Z_1$ and $Z_2$ are two non zero complex numbers such that $\left|Z_1+Z_2\right|=\left|Z_1\right|=\left|Z_2\right|$ then $\frac{Z_1}{Z_2}$ can be equal to
where $\omega$ is the non real cubs root of unity.

• A. $1+\omega$
• B. $1+\omega^2$
• C. $\omega$
• D. $\omega^2$

Answer 1

Answer: (C)

$\text { Let } \frac{ Z _2}{ Z _1}= t \Rightarrow Z _2= Z _1 t , t \neq 0 $
$\therefore \left| Z _1(1+ t )\right|=\left| Z _1\right|=\left| Z _1 t \right| $
$|1+ t |=1=| t | \Rightarrow t \cdot \overline{ t }=1$
$\text { now } $ $(1+ t )(1+\overline{ t })= t \overline{ t } $
$1+\overline{ t }+ t + t \overline{ t }= t \overline{ t } $
$\therefore 1+ t +\overline{ t }=0$
$ 1+ t +\frac{1}{ t }=0 \Rightarrow t ^2+ t +1=0 $
$t =\omega \text { or } \omega^2 \Rightarrow C , D$