If a line through $\mathrm{P}(-2,3)$ meets the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}+2\mathrm{y}+\mathrm{k}=0$ at A and B such that PA.PB = 31 then the radius of the circle is

$\text{ If a secant line through }\mathrm{P}\text{ cuts the circle in any two }$$\text{ points }A,B\text{ then PA.PB is called as power of }P\text{. }$

$\mathrm{PA}\cdot \mathrm{PB}=\left|{\mathrm{S}}_{11}\right|=31$

$\mathrm{S}\equiv {\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}+2\mathrm{y}+\mathrm{k}=0$

$\left|S_{11}\right|={\left(-2\right)}^2+3^2-4\left(-2\right)+2\left(3\right)+k=31\Rightarrow k=4$

Equation of circle is $\mathrm{S}\equiv {\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}+2\mathrm{y}+4=0$

$\mathrm{radius}=\sqrt{4+1-4}=1$