Question:
Show that the equation $x^{2}+y^{2}-3 x+3 y+10=0$ does not represent a circle.
Solution:
Radius = $\sqrt{g^{2}+f^{2}-c}$
$=\sqrt{\left(-\frac{3}{2}\right)^{2}+\left(-\frac{3^{2}}{2}\right)-10}$
$=\sqrt{\frac{9}{2}-10}=\sqrt{-\frac{11}{2}}$
which implies that the radius is negative. (not possible)
Therefore, $x^{2}+y^{2}-3 x+3 y+10=0$ does not represent a circle.