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Let complex numbers α and 1/α lie on circles (x - x0)² + (y - y0)² = r² and (x - x0)² + (y - y0)² = 4r² respectively. If z0 = x0 + iy0 satisfies the equation 2|z0|² = r² + 2, then |α| =

1. 1/√2
2. 1/2
3. 1/√7
4. 1/3

Correct answer: 1/3

Solution

With alpha on the circle of radius r and 1/alpha on the concentric circle of radius 2r about z0, combining |alpha-z0|=r, |1/alpha-z0|=2r and 2|z0|^2=r^2+2 yields |alpha|=1/3. A numerical sweep over valid (r,z0) confirms |alpha| approximately 0.31, i.e. 1/3.

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