Question

Let a complex number $z ,|z| \neq 1$, satisfy $\log _{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^{2}}\right) \leq 2$. Then, the largest value of $|z|$ is equal to

• A. 8
• B. 7
• C. 6
• D. 5

Answer 1

Answer: (B)

$\log _{\frac{1}{\sqrt{2}}}\left(\frac{| z |+11}{( I z \mid-1)^{2}}\right) \leq 2$
$\frac{|z|+11}{(|z|-1)^{2}} \geq \frac{1}{2}$
$2|z|+22 \geq(|z|-1)^{2}$
$2|z|+22 \geq|z|^{2}+1-2|z|$
$|z|^{2}-4|z|-21 \leq 0$
$\Rightarrow \quad|z| \leq 7$
$\therefore $ Largest value of $|z|$ is $7$