Let a complex number z, |z| [] 1, satisfy [ _1 2 ( |z| + 11 (|z| - 1)^2 ) 2].

Question

Let a complex number z, |z| $$\ne$$ 1,

satisfy $${\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2$$. Then, the largest value of |z| is equal to \_\_\_\_\_\_\_\_\_\_\_\_.

Accepted Answer

$${{|z| + 11} \over {{{(|z| - 1)}^2}}} \ge {1 \over 2}$$

$$2|z| + 22 \ge {(|z| - 1)^2}$$

$$2|z| + 22 \ge \,|z{|^2} - 2|z| + 1$$

$$|z{|^2} - 4|z| - 21 \le 0$$

$$(|z| - 7)(|z| + 3) \le 0$$

$$ \Rightarrow \,|z| \le 7$$

$$ \therefore $$ $$|z{|_{\max }} = 7$$

Solutions · 1