Let Z be a complex number with non-zero imaginary part. If 2+3z+4z22−3z+4z2 is a real number, then the value of 10|z|2 =_____

Let X=4z2+3z+24z2−3z+2=1+6z4z2−3z+2=1+62(2z+1z)−3 2z+1z=2z¯+1z¯⇒(z−z¯)=z−z¯|z|2(z−z¯)(2−1|z|2)=0|z|2=12

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Let Z be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z^{2}}{2 - 3 z + 4 z^{2}}$ is a real number, then the value of ${10 | z |}^{2}$ =_____

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answer is 5.

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Detailed Solution

$L e t X = \frac{4 z^{2} + 3 z + 2}{4 z^{2} - 3 z + 2} = 1 + \frac{6 z}{4 z^{2} - 3 z + 2} = 1 + \frac{6}{2 (2 z + \frac{1}{z}) - 3} \begin{array}{l} 2 z + \frac{1}{z} = 2 \bar{z} + \frac{1}{\bar{z}} \Rightarrow (z - \bar{z}) = \frac{z - \bar{z}}{{| z |}^{2}} \\ (z - \bar{z}) (2 - \frac{1}{{| z |}^{2}}) = 0 \\ {| z |}^{2} = \frac{1}{2} \end{array}$