Question:medium

Let z = x+iy represent a point P(x, y) in the Argand plane. If z satisfies the condition that amplitude of $ \frac{z-3}{z-2i} = \frac{\pi}{2} $, then the locus of P is

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The locus $\arg\left(\frac{z-z_1}{z-z_2}\right) = \theta$ represents an arc of a circle passing through points $z_1$ and $z_2$. If $\theta = \pm \pi/2$, it's a semicircle with the segment $z_1z_2$ as diameter. The sign of $\theta$ determines which of the two semicircles is the locus.

Updated On: Mar 30, 2026

the circle $x^2 + y^2 - 3x - 2y = 0$
the arc of the circle $x^2+y^2-3x-2y=0$ intercepted by the diameter $2x+3y-6=0$ containing the origin and excluding the points (3,0) and (0,2)
the arc of the circle $x^2+y^2-3x-2y=0$ intercepted by the diameter $2x+3y-6=0$ not containing the origin and excluding the points (3,0) and (0,2)
the circle $x^2+y^2-3x-2y=0$ not containing the point (0,2)

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