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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 \(\frac{a}{b}\).

 May 2, 2023
 #1
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Since ∣a∣=∣b∣=∣a+b∣, we have ∣a∣2=∣b∣2=∣a+b∣2. This gives us the following equations:

a^2 = b^2 a^2 + b^2 = (a + b)^2

The first equation tells us that a and b are of equal magnitude. The second equation tells us that a2+b2=a2+2ab+b2, or 2ab=0. This means that a and b are either equal or opposite in sign.

If a and b are equal, then a/b=1. If a and b are opposite in sign, then a/b=−1. The sum of all possible values of a/b is therefore 1−1=0​.

 May 2, 2023

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