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Let \(a\) and \(b \) be nonzero complex numbers such that \(|a| = |b| = |a + b|\). Find the sum of all possible values of \(\frac{a}{b}\).

 Jun 29, 2019
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\(\text{To start let's assume $a=1,~b=e^{i\theta}$}\\ a+b = 1+\cos(\theta)+i \sin(\theta)\\ |a+b| = (1+\cos(\theta))^2 + \sin^2(\theta) = 2 + 2 \cos(\theta) = 1\\ \cos(\theta) = -\dfrac 1 2\\ \theta = \pm \dfrac{2\pi}{3}\)

 

\(\text{So now we can let $a=Me^{i\phi},~b = Me^{i (\phi\pm 2\pi/3)}$}\\ \text{and from the previous result we know $|a+b|=M$} \)

 

\(\dfrac a b = \dfrac{Me^{i\phi}}{Me^{i(\phi \pm 2\pi/3)}} = e^{i\pm 2\pi/3}\\ e^{i2\pi/3}+e^{-i 2\pi/3} = 2\cos\left(\dfrac{2\pi}{3}\right) = -1\)

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 Jun 29, 2019
edited by Rom  Jun 30, 2019

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