Question

If $\omega$ is a complex number stisfying $|\omega +\frac{1}{\omega }|$=2,

then maximum distance of $\omega$ from origin is

• A. 2+$\sqrt{3}$
• B. 1+$\sqrt{2}$
• C. 1+$\sqrt{3}$
• D. None of these

Answer 1

The correct option is B

1+$\sqrt{2}$

Since maximum distance of any complex number $\omega$ from origin is given by |$\omega$|

therefore,

|$\omega$|=$|\omega +\frac{1}{\omega }-\frac{1}{\omega }|$ $\le$ $|\omega +\frac{1}{\omega }|$+$\left|\frac{1}{\omega }\right|$=2+$\frac{1}{|\omega |}$

$\Rightarrow$ $|\omega {|}^2$-2|$\omega$|-1 $\le$0 $\Rightarrow$ $\frac{2\pm 2\sqrt{2}}{2}$

Hence max|$\omega$| is 1+$\sqrt{2}$.