The points representing the complex number z for which $a r g (\frac{z - 2}{z + 2}) = \frac{π}{3}$ lie on:

A circle

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A straight line

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An ellipse

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A parabola

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Solution

The correct option is A A circle
Given that $\frac{a r g (z - 2)}{a r g (z + 2)} = \frac{π}{3}$
$\Rightarrow a r g (z - 2) - a r g (z + 2) = \frac{π}{3}$
Hence, two fixed points $(- 2, 0)$ and $(2, 0)$ subtend an angle of $\frac{π}{3}$ at $z$ .
So, $z$ lies on major arc of a circle whose chord is the line joining $(- 2, 0)$ and $(2, 0)$ as shown in the figure.

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