Indian Statistical Institute ( ISI ) B.Math & B.Stat :Complex Numbers

Let $z$ be a non-zero complex number such that $|z −\frac{1}{z}| = 2.$ What is the maximum value of $|z|$? Given $2 = |z −\frac{1}{z}| \geq \big||z|- |\frac{1}{z}|\big|$ Let $t = |z|$ $\implies \big|t- \frac{1}{t}\big| \leq 2$ $\implies -2 \leq t- \frac{1}{t} \leq 2$ $\implies -2t \leq t^2- 1 \leq 2t$ $\implies t^2 +2t- 1 \geq 0 \quad and \quad t^2 -2t- 1 \leq 0$ The first inequality gives $t \in ( - \infty, -1-\sqrt{2}] \cup [\sqrt{2}-1, \infty)$. Since $t \geq 0$ $\implies t \in [\sqrt{2}-1, \infty)$. The second inequality gives $t \in [1-\sqrt{2} , 1+\sqrt{2}]$. Again since $t \geq 0$ $\implies t \in [0,1+\sqrt{2}]$ Combining the two case we see $t \in [\sqrt{2}-1,\sqrt{2}+1] \implies |z| \in [\sqrt{2}-1,\sqrt{2}+1]$. Thus the maximum value of $|z|$ is $\sqrt{2}+1$. $Practice-Problem$ Let $z$ be a non-zero complex number such that $|z +\frac{1}{z}| = a, a \in \mathbb{R^+}.$ What is the maximum and minimum value of $|z|$?