Question

If $|z+\frac{1}{z}|=a$ where $a$ is real number, then find $a$ so that the greatest and least values of $a$ are equal.

• A. $a=2$
• B. $a=-2$
• C. $a=\pm 2$
• D. $a=\pm 1$

Answer 1

Answer: (C)

$a=\pm 2$

$|z+\frac{1}{z}|\le |z|+\frac{1}{|z|}$
$\Rightarrow a\le |z|+\frac{1}{|z|}$
$\Rightarrow |z{|}^2-a|z|+1\ge 0$
$D\le 0$
$\Rightarrow a^2-4\le 0$
$\Rightarrow |a|\le 2$
$aϵ[-2,2]$.
Now since greatest and the lest value is equal.
$D=0$
$\therefore a=\pm 2$.