If z−$$4z=2$$ then the maximum value of |z| is equal to

|z|$$=z$$−$$4z+4z$$≤z−$$4z+4$$|z||z|≤$$2+4$$|z||z|$$2$$−$$2$$|z|−$$4$$≤$$01$$−$$5$$≤|z|≤$$5+1$$

Algebra of complex numbers

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$If |z - \frac{4}{z}| = 2 then the maximum value of | z | is equal to$

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Solution

$\begin{array}{l} | z | = |(z - \frac{4}{z}) + \frac{4}{z}| \leq |z - \frac{4}{z}| + \frac{4}{| z |} \\ | z | \leq 2 + \frac{4}{| z |} \\ | z |^{2} - 2 | z | - 4 \leq 0 \\ 1 - \sqrt{5} \leq | z | \leq \sqrt{5} + 1 \end{array}$

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