if |z-4/z| =2 then what is the maximum value of |z| is equal to - Maths - Complex Numbers and Quadratic Equations

Question

if |z-4/z| =2 then what is the maximum value of |z| is equal
to

Lalit Mehra · Accepted Answer

$Given, | z - \frac{4}{z} | = 2 ∴ | | z | - \frac{4}{| z |} | \leq 2 \Rightarrow - 2 \leq | z | - \frac{4}{| z |} \leq 2$

$\Rightarrow \frac{| z |^{2} - 4}{| z |} \geq - 2 and \frac{{| z |}^{2} - 4}{| z |} \leq 2 \Rightarrow {| z |}^{2} - 4 \geq - 2 | z | and {| z |}^{2} - 4 \leq 2 | z | \Rightarrow {| z |}^{2} + 2 | z | - 4 \geq 0 and {| z |}^{2} - 2 | z | - 4 \leq 0 \Rightarrow {| z |}^{2} + 2 | z | + 1 \geq 4 + 1 and {| z |}^{2} - 2 | z | + 1 \leq 4 + 1 \Rightarrow (| z | + 1)^{2} \geq 5 and (| z | - 1)^{2} \leq 5 \Rightarrow | z | + 1 \geq \pm \sqrt{5} and | z | - 1 \leq \pm \sqrt{5} \Rightarrow | z | \geq \sqrt{5} - 1 and | z | \leq \sqrt{5} + 1 ∴ Maximum value of | z | = \sqrt{5} + 1$