Find the value of a,b if ${\mathrm{ax}}^2+\mathrm{bxy}+3{\mathrm{y}}^2-5\mathrm{x}+2\mathrm{y}-3=0$ represents a circle. Also find radius and centre of the circle.

Given equation is ${\mathrm{ax}}^2+\mathrm{bxy}+3{\mathrm{y}}^2-5\mathrm{x}+2\mathrm{y}-3=0$
It represents a circle
$\Rightarrow \text{ coeff.of }{\mathrm{x}}^2=\text{ coeff .of }{\mathrm{y}}^2\text{ \& coeff of }\mathrm{xy}=0$
$\Rightarrow \mathrm{a}=3\mathrm{\&}\Rightarrow \mathrm{b}=0$
$\therefore \text{ Equation is }3{\mathrm{x}}^2+3{\mathrm{y}}^2-5\mathrm{x}+2\mathrm{y}-3=0$
$\Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-\frac{5}{3}\mathrm{x}+\frac{2}{3}\mathrm{y}-1=0$
$\text{ Centre }\mathrm{C}=(-\mathrm{g},-\mathrm{f})=\left(\frac{5}{6},\frac{-1}{3}\right)$
$\text{ Radius }\mathrm{r}=\sqrt{{\mathrm{g}}^2+{\mathrm{f}}^2-\mathrm{c}}$
$=\sqrt{\frac{25}{36}+\frac{1}{9}+1}=\frac{\sqrt{65}}{6}$