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Conjugate Complex Numbers

Source: 1995 National High School Mathematics League, Exam One, Problem 7

March 3, 2020
complex numbers

Problem Statement

α,β\alpha,\beta are conjugate complex numbers. If ∣α−β∣=23|\alpha-\beta|=2\sqrt3, αβ2\frac{\alpha}{\beta^2} is a real number, then ∣α∣=|\alpha|=________.
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