The center and radius of the circle $2x^2+2y^2-4x+6y-3=0$

$2x^2+2y^2-4x+6y-3=0$

$\Rightarrow x^2+y^2-2x+3y-\frac{3}{2}=0$

Centre $(-g,-f)=(1,-\frac{3}{2})$

Radius $=\sqrt{g^2+f^2-c}$

$\begin{array}{l}=\sqrt{{(-1)}^2+{\left(\frac{3}{2}\right)}^2+\frac{3}{2}}\\ =\sqrt{\frac{4+9+6}{4}}\\ =\frac{\sqrt{19}}{2}\end{array}$