Find the values of ‘a’ if $2{\mathrm{x}}^2+{\mathrm{ay}}^2-3\mathrm{x}+2\mathrm{y}-1=0$ represents a circle and also find its radius.

Given equation is $2{\mathrm{x}}^2+{\mathrm{ay}}^2-3\mathrm{x}+2\mathrm{y}-1=0$
It represents a circle
$\begin{array}{l}\Rightarrow \text{ coefficient of }{\mathrm{x}}^2=\text{ coefficient of }{\mathrm{y}}^2\\ \Rightarrow 2=\mathrm{a}\therefore \mathrm{a}=2\end{array}$
The equation becomes
$\begin{array}{l}2{\mathrm{x}}^2+2{\mathrm{y}}^2-3\mathrm{x}+2\mathrm{y}-1=0\\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-\frac{3}{2}\mathrm{x}+\mathrm{y}-\frac{1}{2}=0(\text{ in its. form) }\\ \mathrm{g}=\frac{-3}{4},\mathrm{f}=\frac{1}{2},\mathrm{c}=\frac{-1}{2}\end{array}$
$\text{ Radius }\mathrm{r}=\sqrt{{\mathrm{g}}^2+{\mathrm{f}}^2-\mathrm{c}}$
$=\sqrt{\frac{9}{16}+\frac{1}{4}+\frac{1}{2}}=\sqrt{\frac{9+4+8}{16}}=\frac{\sqrt{21}}{4}$