The equation of ellipse is $\frac{x^2}{9}+y^2=1$

The length of semi-major axis is $a =3$ and the length of semi-minor axis is $b =1$

The coordinates of point $A$ is $(3,0)$ and the coordinates of point $B$ is $(0,1)$

The equation of line passing through $A , B$ is $x +3 y =3$

The equation of an auxiliary circle of an ellipse is $x^2+y^2=9$

The line $A B$ cuts $x^2+y^2=9$ at point $M$

By solving the above equations

The coordinates of point $M$ are $\left(-\frac{12}{5}, \frac{9}{5}\right)$

The area of triangle AMO is $\frac{1}{2}\left|0\left(0-\frac{9}{5}\right)+3\left(\frac{9}{5}-0\right)-\frac{12}{5}(0-0)\right|=\frac{27}{10}$