$\sqrt{\frac{(x-1)^{2}+y^{2}}{x^{2}+(y+1)^{2}}}=2$
$x^{2}+y^{2}+\frac{2}{3} x+\frac{8}{3} y+1=0,$ centre $\equiv\left(-\frac{1}{3},-\frac{4}{3}\right)$
radius $=\frac{2 \sqrt{2}}{3}$
Co-ordinates of
$P(\alpha, \beta) \equiv\left(-\frac{1}{3}+\frac{2 \sqrt{2}}{3} \cos \theta,-\frac{4}{3}+\frac{2 \sqrt{2}}{3} \sin \theta\right)$
$(\alpha+\beta)=-\frac{5}{3}+\frac{2 \sqrt{2}}{3}(\cos \theta+\sin \theta)$
$\left[\frac{(\alpha+\beta)_{\max }}{(\alpha+\beta)_{\min }}\right]=\left[\frac{-\frac{1}{3}}{-3}\right]=0$
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Statement $1:$ If arg $Z+$ arg $W = \pi ,$ then $Z = -\overline W $.
Statement $2:$ $\left| Z \right| = \left| W \right|,$ implies arg $Z-$ arg $\overline W = \pi .$