Q.Find the values of $a$ and $b$ if the equation $(a-1)x^2+by^2+(b-8)xy+4x+4y-1=0$ represents a circle.

Expanding the centre-radius form $(x-h)^2+(y-k)^2=r^2$ and matching coefficients with $x^2+y^2+2gx+2fy+c=0$ shows that every circle can be written in this "general" second-degree form, with centre $(-g,-f)$ and radius $\sqrt{g^2+f^2-c}$. Conversely, not every equation shaped like $x^2+y^2+2gx+2fy+c=0$ represents a real circle: the value of $g^2+f^2-c$ decides which of three cases holds — positive (a genuine circle with a positive radius), zero (the equation collapses to a single point, a degenerate circle of radius $0$), or negative (no real point satisfies the equation at all, so no circle exists in the plane). This form is the natural target whenever a circle is defined only indirectly — e.g. by passing through several given …