Question

$z_1$ and $z_2$ are two distinct points in an Argand plane. If $a|z_1|=b|z_2|$ (where a, b$ϵ$ 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 imaginary axis
• C. Unit circle $|z|=1$
• D. The line with arg z=$ta{n}^{-1}2$

Answer 1

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

Let us assume that,
$arg\left(z_1\right)=\theta andarg\left(z_2\right)=\theta +\alpha$
Now,$\frac{a{z}_1}{b{z}_2}=\frac{a|z_1|e^{i\theta }}{b|z_2|e^{i(\theta +\alpha )}}=e^{-i\alpha }$
and,$\frac{b{z}_2}{a{z}_1}=\frac{b|z_2|e^{i(\theta +\alpha )}}{a|z_1|e^{i\theta }}=e^{i\alpha }$
Thus,$\frac{a{z}_1}{b{z}_2}+\frac{b{z}_2}{a{z}_1}=e^{-i\alpha }+e^{i\alpha }=2\cos \left(\alpha \right)$
Hence the point lies on $[-2,2]$