An algebra problem by abhi jith

Algebra Level pending

Let \(z_1\) and \(z_2\) be complex numbers satisfying \(z + \bar z = 2|z-1| \) and \( \arg(z_1 - z_2 ) = \frac \pi4\).

Which of the following statements are true?

A: \(\ \Re(z_1+z_2)=2\)
B: \(\ \Im(z_1+z_2)=2 \)
C: \(\ z_1\) and \(z_2\) lies on a parabola.
D: \(\ z_1\) and \(z_2\) does not exist.

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