The equation of the ellipse whose distance between the foci is $8$ and the distance between the directrices is $18$,is

Let $A_1, A_2, A_3$ be regions in the $XY$-plane defined by:
$A_1 = \{(x, y) : x^2 + 2y^2 \leq 1\}$
$A_2 = \{(x, y) : |x|^3 + 2\sqrt{2}|y|^3 \leq 1\}$
$A_3 = \{(x, y) : \max(|x|, \sqrt{2}|y|) \leq 1\}$
Then,

Let $S$ and $S^{\prime}$ be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $P(\alpha, \beta)$ be a point on the ellipse in the first quadrant. If $(SP)^2+(S^{\prime}P)^2-SP \cdot S^{\prime}P=37$,then $\alpha^2+\beta^2$ is equal to:

If the line $\alpha x+4y=\sqrt{7}$,where $\alpha \in R$,touches the ellipse $3x^{2}+4y^{2}=1$ at the point $P$ in the first quadrant,then one of the focal distances of $P$ is:

The equation of the ellipse having a vertex at $(6,1)$,a focus at $(4,1)$ and the eccentricity $e = \frac{3}{5}$ is

The eccentricity of the ellipse $x^2+4 y^2+2 x+16 y+13=0$ is