Let $\mathrm{S}=\{\mathrm{z} \in \mathrm{C},|\mathrm{z}-1|=1$ and $(\sqrt{2}-1)(\mathrm{z}+\overline{\mathrm{z}})-\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}})=2 \sqrt{2}\}$. Let $\mathrm{z}_{1}, \mathrm{z}_{2} \in \mathrm{~S}$ be such that $\left|\mathrm{z}_{1}\right|=\max _{\mathrm{Z} \in \mathrm{S}}|\mathrm{z}|$ and $\left|\mathrm{z}_{2}\right|=\min _{\mathrm{Z} \in \mathrm{S}}|\mathrm{z}|$. Then $\left|\sqrt{2} \mathrm{z}_{1}-\mathrm{z}_{2}\right|^{2}$ equals

\[ \begin{aligned} & (\sqrt{2}-1)(\mathrm{z}+\overline{\mathrm{z}})-\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}})=2 \sqrt{2} \Rightarrow(\sqrt{2}-1)(2 \mathrm{x})-\mathrm{i}(2 \mathrm{iy})=2 \sqrt{2} \\ & \Rightarrow(\sqrt{2}-1) \mathrm{x}+\mathrm{y}=\sqrt{2} \Rightarrow \mathrm{y}=\sqrt{2}-(\sqrt{2}-1) \mathrm{x} \ldots \text { (ii) On solving with (i) } \\ & (\mathrm{x}-1)^{2}+2+(3-2 \sqrt{2}) \mathrm{x}^{2}-2 \sqrt{2}(\sqrt{2}-1) \mathrm{x}=1 \Rightarrow(2-\sqrt{2}) \mathrm{x}^{2}+(\sqrt{2}-3) \mathrm{x}+1=0 \text { For } \mathrm{x}=1, \mathrm{y}=1, \mathrm{z}=1+\mathrm{i} \text { for }|\mathrm{z}|=\sqrt{2} \text { (least modulus) } \end{aligned} \]