If z is complex number then interpret the locus of following in complex plane $A r g (z + i) - A r g (z - i) = \frac{π}{2}$ .

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Solution

$A r g (z + i) - A r g (z - i) = \frac{Π}{2}$
Let $z = x + i y$
$\Rightarrow A r g (x + i y + i) - A r g (x + i y - i) = Π / 2$
$\Rightarrow \frac{y + 1}{x} - \frac{y - 1}{x} = \frac{1}{0}$
$\Rightarrow \frac{y + 1 - y + 1}{x} = \frac{1}{0}$
$\Rightarrow x = 0$

$∴ x = 0$ is the locus of $z$ .

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