$\begin{array}{l} 3\left|\mathrm{z}_{1}\right|=4\left|\mathrm{z}_{2}\right| \\ \\ \Rightarrow \frac{\left|\mathrm{z}_{1}\right|}{\left|\mathrm{z}_{2}\right|}=\frac{4}{3} \\ \\ \Rightarrow \frac{\left|3 \mathrm{z}_{1}\right|}{\left|2 \mathrm{z}_{2}\right|}=2 \\ \\ \text { Let } \frac{3 \mathrm{z}_{1}}{2 \mathrm{z}_{2}}=\mathrm{a}=2 \cos \theta+2 \mathrm{i} \sin \theta \\ \\ \mathrm{z}=\frac{3 \mathrm{z}_{1}}{2 \mathrm{z}_{2}}+\frac{2 \mathrm{z}_{2}}{3 \mathrm{z}_{1}}=\mathrm{a}+\frac{1}{\mathrm{a}} \\ \\ =\frac{5}{2} \cos \theta+\frac{3}{2} \mathrm{i} \sin \theta \\ \\ |\mathrm{z}|=\sqrt{\frac{17}{2}} \end{array}$

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