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Let z = x + iy be a variable complex number. If arg((z - 1)/(z + 1)) = π/4, then which relation is satisfied?

1. x² − y² − 2x = 1
2. x² + y² − 2x = 1
3. x² + y² − 2y = 1
4. x² + y² + 2x = 1

Correct answer: x² + y² − 2y = 1

Solution

Setting z = x+iy and equating the argument of (z-1)/(z+1) to pi/4 gives an arc of a circle through +1 and -1; algebra (or direct testing of points on the locus) yields x^2 + y^2 - 2y = 1.

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