From page $19$ of Ahlfors' Complex Analysis:
"[...] this equation takes the form [...]
$$(\alpha_0-\alpha_3)(x^2+y^2)-2\alpha_1x-2\alpha_2y+\alpha_0+\alpha_3 = 0.$$
For $\alpha_0\ne \alpha_3$ this is the equation of a cricle, and for $\alpha_0=\alpha_3$ it is the equation of a straight line."
Why does the equation above describe a circle whenever $\alpha_0\ne\alpha_3$ and a line otherwise?
If asked to write (a set of) equations for a circle or a straight line I would either write them as parametrizations or as intersections of a plane and a sphere, or of two planes respectively, yet the equation above does not seem to take any of these forms.
I believe that there's an $x$ missing in your equation (possibly a misprint in the book)
From basic analytic geometry an equation $$ x^2+y^2-2ax-2by+c=0\qquad(\ast) $$ describes a circle of centre $(a,b)$ and radius $r$ with $r^2=a^2+b^2-c$.
If $\alpha_0\neq\alpha_3$ the given equation reduces to $(\ast)$ dividing by a scalar.
If $\alpha_0=\alpha_3$ the quadratic term disappears and you are left with a linear equation which describes a line unless the linear terms disappear as well.