Question

$z_1$ and $z_2$ are two distinct points in an Argand plane. If $a\left|z_1\right|=b\left|z_2\right|$ (where $a,b\in R),$ then the point $\left(a{z}_1/b{z}_2\right)+\left(b{z}_2/a{z}_1\right)$ is a point on the

• A. line segment $[-2,2]$ of the real axis
• B. line segment $[-2,2]$of the imaginaryaxis
• C. unit circle $\left|Z\right|=1$
• D. the line with $argz={\tan }^{-1}2$

Answer 1

The correct option is A line segment $[-2,2]$ of the real axis

Let $z_1=|z_1|e^{i\theta }$ $z_2=|z_2|e^{i(\theta +\alpha )}$

$\frac{a{z}_1}{b{z}_2}$$+\frac{b{z}_2}{a{z}_1}$

$⟹\frac{a|z_1|e^{i\theta }}{b|z_2|e^{i(\theta +\alpha )}}$$\frac{b|z_2|e^{i(\theta +\alpha )}}{a|z_1|e^{i\left(\theta \right)}}$ $(a|z_1|=b|z_2|)$

$⟹e^{-i\alpha }$$+e^{i\alpha }$

$⟹$ Real no. on $[-2,2]$ segments.