4. Given $\odot C: x^{2}+y^{2}-2 x-2 y=-1, A(2 a, 0), B(0,2 b), a>1, b>1$, when $\odot C$ is tangent to the line $A B$, the minimum value of the area of $\triangle A O B$ is $\qquad$ ( $O$ is the origin).

Transform the above text into English, preserving the original text's line breaks and format, and output the translation result directly.
4. $3+2 \sqrt{2}$ Hint: The equation of $AB$ is $b x+a y-2 a b=0$, and $\frac{|b+a-2 a b|}{\sqrt{a^{2}+b^{2}}}=1$, i.e., $2 a b \geqslant 4 \sqrt{a b}-1, \sqrt{a b}$ $\geqslant \frac{2+\sqrt{2}}{2}, S_{\triangle A O B}=2 a b \geqslant 3+2 \sqrt{2}$.