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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 \(\frac{a}{b}.\)

 Mar 1, 2020
edited by CCjump21  Mar 1, 2020
edited by CCjump21  Mar 2, 2020
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The sum of all possible values of a/b is 1 + sqrt(3)*i/2 + 1 - sqrt(3)*i/2 = 2.

 Mar 3, 2020
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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 \(\dfrac{a}{b}\).

 

My attempt:

 

\(\begin{array}{|c|c|c|} \hline a & b & \dfrac{a}{b} \\ \hline 1+\sqrt{3}i & 1-\sqrt{3}i & \dfrac{1+\sqrt{3}i}{ 1-\sqrt{3}i} = -\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i \\ 1-\sqrt{3}i & 1+\sqrt{3}i & \dfrac{ 1-\sqrt{3}i}{1+\sqrt{3}i} = -\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i \\ \hline \sqrt{3}+ i & -\sqrt{3}+i & \dfrac{\sqrt{3}+ i}{-\sqrt{3}+i} = -\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i \\ -\sqrt{3}+i & \sqrt{3}+ i & \dfrac{-\sqrt{3}+i}{\sqrt{3}+ i} = -\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i \\ \hline -1+\sqrt{3}i & -1-\sqrt{3}i & \dfrac{-1+\sqrt{3}i}{ -1-\sqrt{3}i} = -\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i \\ -1-\sqrt{3}i & -1+\sqrt{3}i & \dfrac{ -1-\sqrt{3}i}{-1+\sqrt{3}i} = -\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i \\ \hline \sqrt{3}- i & -\sqrt{3}-i & \dfrac{\sqrt{3}- i}{-\sqrt{3}-i} = -\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i \\ -\sqrt{3}-i & \sqrt{3}- i & \dfrac{-\sqrt{3}-i}{\sqrt{3}- i} = -\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i \\ \hline & & \text{sum}~= -\dfrac{1}{2} -\dfrac{1}{2} -\dfrac{1}{2} -\dfrac{1}{2} -\dfrac{1}{2} -\dfrac{1}{2} -\dfrac{1}{2} -\dfrac{1}{2} \\ & & \text{sum}~= \mathbf{ -4 } \\ \hline \end{array}\)

 

laugh

 Mar 3, 2020

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