Mathematics - Ellipse Question with Solution | TestHub

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 $QR$ and $PQ=\frac{2\sqrt{2}}{3}$, the value of $9\left(x_1{y}_1+x_2{y}_2+x_3{y}_3\right)$ is equal to ______ .