jmc

First, we factor out the constants of the squared terms to get $9(x^2-2x)+9(y^2+4y)=-44.$

To complete the square, we need to add ${\left(\frac{2}{2}\right)}^2=1$ after the $-2x$ and ${\left(\frac{4}{2}\right)}^2=4$ after the $4y,$ giving $$9(x-1{)}^2+9(y+2{)}^2=-44+9+36=1.$$ Dividing the equation by $9$ gives $$(x-1{)}^2+(y+2{)}^2=\frac{1}{9},$$ so the radius is $\sqrt{\frac{1}{9}}=\boxed{\frac{1}{3}}.$