Question:
Let a complex number $\mathrm{z},|\mathrm{z}| \neq 1$,
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______.
Correct Option:
Solution:
$\log _{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^{2}}\right) \leq 2$
$\frac{|z|+11}{(|z|-1)^{2}} \geq \frac{1}{2}$
$2|z|+22 \geq(|z|-1)^{2}$
$2|z|+22 \geq|z|^{2}+1-2|z|$
$|z|^{2}-4|z|-21 \leq 0$
$\Rightarrow|z| \leq 7$
$\therefore \quad$ Largest value of $|z|$ is 7
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