Question

The locus of the complex number arg(z^* + i√3) - π/4 is equal to?

Answer 1

$\text{Given $\rm{arg }(z + i \sqrt{3}) = \frac\pi4$}\\ \text{Note that : $\rm{arg }(z + i \sqrt{3}) = \rm{arg }(x + i(y+ \sqrt{3})) = \tan^{-1} \frac{y + \sqrt3}{x} = \frac\pi4$} \\ \implies \frac{y + \sqrt3}{x} = \tan \frac\pi4 = 1 \\ \implies x = y + \sqrt3 \Leftrightarrow y = x - \sqrt3 \\ \text{$\therefore$ the locus of the given equation is } y = x - \sqrt3$