Question

Let $z$ be a complex number such that $|z+\frac{1}{z}|=2$.

If $\left|z\right|=r_1$ and $r_2$ for $\arg z=\frac{\pi }{4}$ then

$\arg z$ varies $|r_1-r_2|=$

• A. $[0,2]$
• B. $|0,1|$
• C. $[\frac{1}{\sqrt{2}},\sqrt{2}]$
• D. $|1,2|$

Answer 1

The correct option is A $[0,2]$

Ans. $\left(b\right)$.
If $z_1=r_1{e}^{i{\theta }_1}$, $z_2=r_2{e}^{i{\theta }_2}$, then
${|z_1+z_2|}^2=r_1^2+r_2^2+2r_1{r}_2\cos ({\theta }_1-{\theta }_2)$
If $z=r{e}^{1{\theta }_1},\frac{1}{z}=\frac{1}{r}{e}^{-i\theta }$
Ans. $\left(a\right)$.
As $\theta$ varies, max. value of $\sin {}^2\theta =1$ and min. is $0$.
$\therefore {(r-\frac{1}{r})}^2$ varies from $0$ to $4$.
or $|r_1-r_2|\in [0,2]$.