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Q.
If $3^{50}(x + iy) = \left(\frac{3}{2} + \frac{i \sqrt{3}}{2} \right)^{100} \,\forall \,x, y \in R$, then ordered pair $(x, y)$ is given by
Complex Numbers and Quadratic Equations
Solution:
The given equation can be written as
$\left(i\sqrt{3}\right)^{100} \left(\frac{1}{2} - \frac{i\sqrt{3}}{2}\right)^{100} = 3^{50}\left(x + iy\right) $
$ \Rightarrow \left(-\omega\right)^{100} = x + iy $
$ \Rightarrow \omega = x + iy $
$ \Rightarrow \frac{-1 + i\sqrt{3}}{2} = x + iy $
$\Rightarrow x = -1/2, y= \frac{\sqrt{3}}{2}$