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Equation mixing |z| and z: compare real and imaginary parts

This type of question, asked 2 times: 8 Apr 2023, 29 Jan 2024. All 2 questions below, oldest first.

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  1. Question 1

    8 Apr 2023

    80
    If for z=α+iβ,∣z+2∣=z+4(1+i)z = \alpha + i\beta,|z + 2| = z + 4(1 + i), then α+β\alpha + \beta and αβ\alpha\beta are the roots of the equation

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  2. Question 2

    29 Jan 2024

    65
    If z=12−2iz =\frac{1}{2}- 2i, is such that ∣z+1∣=αz+β(1+i),i=−1|z + 1| = \alpha z + \beta(1 + i),i =\sqrt{- 1} and  α,β∈R ,\ \alpha,\beta \in R\ , then α+β\alpha + \beta is equal to

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