Question

Complex number $z$ satisfy the equation $|z-(4/z\left)\right|=2$. Then the value of $arg\left(z_1/z_2\right)$, where $z_1$ and $z_2$ are complex numbers with the greatest and the least moduli, can be

• A. 2$\pi$
• B. $\pi$
• C. $\frac{\pi }{2}$
• D. none of these

Answer 1

The correct option is B $\pi$

Re-framing the equation, we get
$|z{|}^2-4=2|z|$
$(|z|-1{)}^2=5$
And similarly $(|z|+1{)}^2=5$
$|z{|}_{max}=\sqrt{5}+1$ and $|z{|}_{min}=\sqrt{5}-1$
$\frac{|z{|}_{max}}{|z{|}_{min}}$
$=\frac{(\sqrt{5}+1{)}^2}{5-1}$
$=\frac{6+2\sqrt{5}}{4}$
$=\frac{3+\sqrt{5}}{2}$
$\frac{z_1}{z_2}$
$=\frac{r_1{e}^{ia}}{r_2{e}^{ib}}$
$=\frac{3+\sqrt{5}}{2}.e^{i(a-b)}$
$a-b\ne 0,2\pi$ otherwise $z_1$ and $z_2$ would be equal.
$a=\pi +b$
$arg\frac{z_1}{z_2}$ can be $\pi$
Hence, option 'B' is correct.