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Algebra Level 5

Consider two complex numbers: \(z_1\) : \(692+3745\sqrt{3}i\) and \(z_2\) : \(692+1729\sqrt{3}i\).

There exists another complex number \(z\) such that \(\text{arg}\left(\dfrac{z-z_1}{z-z_2}\right)=\dfrac{\pi}{3}\).

Then , Find the value of \(|z+316-2737\sqrt{3}i|\)

Details And Assumptions:

  • \(\text{arg}(z)\) is the argument which means the angle made by the complex number \(z\) with the positive X axis.
  • \(|z|\) represents the modulus of complex number \(z\).
  • \(i=\sqrt{-1}\)


This problem is original.

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