Ellipse question bank (Maths)

Let $\dfrac{x^2}{f(a^2+7a+3)} + \dfrac{y^2}{f(3a+15)} = 1$ represent an ellipse with major axis along $y$-axis, where $f$ is a strictly decreasing positive function on $\mathbb{R}$. If the set of all possible values of $a$ is $\mathbb{R} - [\alpha, \beta]$, then $\alpha^2+\beta^2$ is equal to:

Let $x = 9$ be a directrix of an ellipse $E$, whose centre is at the origin and eccentricity is $\dfrac{1}{3}$. Let $P(\alpha, 0)$, $\alpha \gt 0$, be a focus of $E$ and $AB$ be a chord passing through $P$. Then the locus of the mid point of $AB$ is :