Let C be the circle x^2+(y-1)^2=2, E_1 and E_2 be two ellipses...

Let C be the circle $\mathrm{x}^2+(\mathrm{y}-1)^2=2, \mathrm{E}_1$ and $\mathrm{E}_2$ be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line $x+y=3$ touch the curves $C$, $E_1$ and $E_2$ at $P$\left(x_1, y_1\right), $Q\left(x_2, y_2\right)$ and $$R\left(x_3, y_3\right)$ respectively. Given that $P$ is the mid-point of the line segment $Q R$ and $P Q=\frac{2 \sqrt{2}}{3}$, the value of $9$\left(x_1 y_1+x_2 y_2+x_3 y_3\right)$ is equal to ______ .