$\text{ If }a\left(x^2+y^2+2y+1\right)=(x-2y+3{)}^2\text{ is an ellipse and }a\in (b,\mathrm{\infty }),\text{ then the value of }b\text{ is }$

$\begin{array}{l}a\left(x^2+y^2+2y+1\right)=(x-2y+3{)}^2\\ \Rightarrow a\left(x^2+(y+1{)}^2\right)=(x-2y+3{)}^2\\ \Rightarrow \sqrt{x^2+(y+1{)}^2}=\frac{\sqrt{5}}{\sqrt{a}}\frac{|x-2y+3|}{\sqrt{5}}\\ \text{ This represents ellipse if }e=\frac{\sqrt{5}}{\sqrt{a}}<1\text{ or }a\in (5,\mathrm{\infty })\end{array}$