If z is complex number then interpret the locus of following in complex plane $A r g (z - a) = \frac{π}{3}$ where $a = 3 + 4 i$ .

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Solution

Let $z = x + i y$
$\Rightarrow A r g (x + i y - 3 - 4 i) = Π / 3$
$\Rightarrow (A r g (x - 3) + (y - 4) i) = Π / 3$
$\Rightarrow \frac{y - 4}{x - 3} = \sqrt{3}$
$\Rightarrow y - 4 = \sqrt{3} x - 3 \sqrt{3}$
$\Rightarrow \sqrt{3} x - y = 3 \sqrt{3} - 4$ .

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