Let r 1 and r 2 be the radii of the largest and smallest circles, respectively, which pass through the point…

We have,

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

$\Rightarrow x^2+2x+1+y^2+4y+4=9$

$\Rightarrow {\left(x+1\right)}^2+{\left(y+2\right)}^2=3^2$

Centre$\equiv \left(-1,-2\right)$

$CP=\sqrt{{\left(-4+1\right)}^2+{\left(1+2\right)}^2}=3\sqrt{2}$

Centre of smallest circle is $A$

Centre of largest circle is $B$.

$r_2=|CP-CA|=3\sqrt{2}-3$

$r_1=CP+CB=3\sqrt{2}+3$

Hence,

$\frac{r_1}{r_2}=\frac{3\sqrt{2}+3}{3\sqrt{2}-3}=\frac{(3\sqrt{2}+3{)}^2}{9}$

$\Rightarrow \frac{r_1}{r_2}=(\sqrt{2}+1{)}^2=3+2\sqrt{2}$

Hence,

$a=3, b=2$

$a+b=5$