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Let and be nonzero complex numbers such that . Find the sum of all possible values of .

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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