if |z-(3+4i)|<=3 then find the complex number having least magnitude satisfying the above inequality

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that}: \\ \left|\mathrm{z}-{\mathrm{z}}_1\right|=\mathrm{r} \mathrm{represents} \mathrm{a} \mathrm{circle} \mathrm{of} \mathrm{radius} \mathrm{of} \mathrm{r} \mathrm{and} \mathrm{centre} {\mathrm{z}}_1 \\ \mathrm{Hence} \\ \left|\mathrm{z}-\left(3+4\mathrm{i}\right)\right|\le 3 \mathrm{will} \mathrm{represent} \mathrm{a} \mathrm{interior} \mathrm{and} \mathrm{boundary} \mathrm{of} \mathrm{circle} \mathrm{of} \mathrm{radius} 3 \mathrm{and} \\ \mathrm{centre} \left(3+4\mathrm{i}\right) \\ \mathrm{Remember} \mathrm{that} \mathrm{least} \mathrm{and} \mathrm{maximum} \mathrm{distance} \mathrm{both} \mathrm{are} \mathrm{obtained} \mathrm{along} \mathrm{line} \\ \mathrm{joining} \mathrm{centre} \mathrm{and} \mathrm{origin}. \\ \mathrm{Distance} \mathrm{between} \mathrm{origin} \mathrm{and} \mathrm{centre}=\left|3+4\mathrm{i}\right|=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5 \\ \mathrm{Radius}=3 \\ {\left|\mathrm{z}\right|}_{\min }=5-3=\mathbf{2}\mathbf{ }\left(\mathbf{Answer}\right) \\ \mathrm{Similarly} \\ {\mathrm{z}}_{\max }=5+3=8 \end{gathered}$$

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