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Among all complex numbers z satisfying |z + 3 - sqrt(3) i| = sqrt(3), find the one having the least positive argument.

1. -3/2 + (sqrt(3)/2) i
2. -3 + 0 i
3. 0 + sqrt(3) i
4. -3/2 + (3*sqrt(3)/2) i

Correct answer: -3/2 + (3*sqrt(3)/2) i

Solution

The centre is at distance 2*sqrt(3) from the origin (radius sqrt(3)), so the half-angle is 30 deg and the tangent of least positive argument is at 120 deg. The point of tangency, at distance 3 from the origin, is -3/2 + (3*sqrt(3)/2) i.

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