.Complex Nos.

Algebra Level 4

Given: \[ |3z_1 -2z_2-4|^2 = |3z_1 -1|^2 + |2z_2 + 3|^2 ;\left(z_2 \neq -\dfrac{3}{2} \right). \]
Cube Roots of \(\omega =\dfrac{3z_1-1}{2z_2+3} \) are \(\omega_1,\omega_2,\omega_3\); Also \(\text{arg}\omega_1 <\text{arg}\omega_2<\text{arg}\omega_3 \).
Then,
\[ \dfrac{\omega_{2} ^2}{\omega_1 \omega_3}\] would be?

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