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Real numbers a and b each lie between 0 and 1 and are such that the points z1 = a + i, z2 = 1 + bi and z3 = 0 are the vertices of an equilateral triangle in the complex plane. Find a and b.

1. a = b = 2 - sqrt(3)
2. a = b = 2 + sqrt(3)
3. a = b = sqrt(3) - 1
4. a = b = 1/2

Correct answer: a = b = 2 - sqrt(3)

Solution

Setting the side z1z3 equal to side z1z2 with a = b gives a² - 4a + 1 = 0, whose root in (0,1) is 2 - sqrt(3).

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