Ellipse question bank (Maths)

The eccentricity of an ellipse $E$ with centre at the origin $O$ is $\dfrac{\sqrt{3}}{2}$ and its directrices are $x = \pm \dfrac{4\sqrt{6}}{3}$. Let $H: \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of $E$, and whose length of latus rectum is equal to the length of minor axis of $E$. Then the distance between the foci of $H$ is :

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 ______ .