Question

locate the points representing the complex number $z$ for which
$\arg \left(\frac{z-1-i}{z-2}\right)=\frac{\pi }{3}$

Answer 1

We know from Cor P.71 that
$\arg \left(\frac{z-z_1}{z-z_2}\right)=\alpha$ represents a circle whose chord $AB$ joining the point $z_1$ and $z_2$ subtends an angle $\alpha$ at the circumference. Hence the equation represents a circle whose equation can be shown to be
$\sqrt{3}(x^2+y^2)-(3\sqrt{3}-1)x-(\sqrt{3}-1)y+2\sqrt{3}-2=0$
$\because \tan {}^{-1}\frac{y-1}{x-1}-\tan {}^{-1}\frac{y}{x-2}=\frac{\pi }{3}$ etc.