Find the locus of complex numbers z satisfying the condition $| z + 3 - \sqrt{3} i | = \sqrt{3}$

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Solution

Let $z = x + i y$
$∣ ∣ z + 3 - \sqrt{3} i ∣ ∣ = \sqrt{3}$
$\sqrt{(x + 3)^{2} + (y - \sqrt{3})^{2}} = \sqrt{3}$
$x^{2} + 6 x + 9 + y^{2} - 2 \sqrt{3} y + 3 = 3$
$\Rightarrow x^{2} + y^{2} + 6 x - 2 \sqrt{3} y + 9 = 0$

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