Let C be the circle x^2+(y-1)^2=2, E_1 and E_2 be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively.…

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