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Let z_{1} \:and \:z_{2} be any two non-zero complex numbers such that

 3|z_{1}| = 4|z_{2}|. If z= \frac{3z_{1}}{2z_{2}}+\frac{2z_{2}}{3z_{1}} then :

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It is given that z_{1} \:and \:z_{2}  are two non- zero complex number such that,
3|z_{1}| = 4|z_{2}|

therefore,
z= \frac{3z_{1}}{2z_{2}}+\frac{2z_{2}}{3z_{1}}
= \frac{3}{2}(\frac{4}{3}.\frac{z_2}{z_2})+\frac{2.z_2}{3\times (4/3.z_2)}
\\=\frac{3}{2}\times \frac{4}{3}+\frac{1}{2}\\ =\frac{5}{2}

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manish

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