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Complex Plane

Source: 1996 National High School Mathematics League, Exam One, Problem 8

March 3, 2020
complex numbers

Problem Statement

On the complex plane, non-zero complex numbers z1,z2z_1,z_2 are on the circle with center i\text{i}, radius of 11. The real part of z1‾⋅z2\overline{z_1}\cdot z_2 is 00, and arg⁡(z1)=π6\arg (z_1)=\frac{\pi}{6}, then z2=z_2=________.
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