"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 ____________."

Question

Accepted Answer

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

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

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

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

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

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

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

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 ____________.

Solution

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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 ____________.

Solution

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More Complex Numbers and Quadratic Equations questions

Drill 25 more like these. Every day. Free.

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 ____________.