Question

If z be a complex number for which $|2zcos\theta +z^2|=1$, then the minimum value of |z|
is ......................

• A. $\sqrt{3}-1$
• B. $\sqrt{3}+1$
• C. $\sqrt{2}-1$
• D. $\sqrt{2}+1$

Answer 1

The correct option is C $\sqrt{2}-1$

$|z^2+2zcos\theta |$
$=|z(z+2cos\theta )|$
$=|z|.|z+2cos\theta |$
$=1$
Now
$|z|=1$ and
$|z+2cos\theta |=1$
Now
$|z+2cos\theta |\le |z|+|2cos\theta |$
Considering
$|z+2\cos \theta |=|z|+|2cos\theta |=1$
Hence
$|z|=|2cos\theta |\pm 1$
Considering $z=|2cos\theta |-1$ we get the minimum value at multiples of $\theta ={45}^0$
Hence
$z=\sqrt{2}-1$.