Locus In Argand Plane

Question

The set of all complex numbers z satisfying the equation given by the ratio condition arg of the quantity (z - 1)/(z + 1) equal to π/2 forms which curve, excluding endpoints?

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The argument of a ratio (z - a)/(z - b) measures the directed angle subtended at z by the segment from a to b, a powerful JEE Advanced locus tool. Setting arg((z - 1)/(z + 1)) = π/2 means the segment joining the points 1 and -1 subtends a right angle at z. By the converse of the angle-in-semicircle theorem, the locus of points seeing a fixed segment at 90 degrees is the circle whose diameter is that segment. The segment from -1 to 1 has midpoint at the origin and length 2, so the circle has centre 0 and radius 1. The fixed sign of the argument restricts z to one side, giving a semicircle, with the diameter endpoints excluded since the ratio is undefined there. Option full circle ignores the sign restriction. Option straight line would arise if the argument were 0 or π. Option parabola has no basis. Hence the locus is a semicircle of radius 1 centred at the origin. Plausibility check: the point z = i gives (i-1)/(i+1) = i, whose argument is π/2, and |i| = 1, confirming i lies on the described semicircle.

Locus In Argand Plane

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