Let each of the two ellipses $\mathrm{E}_1: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ and $\mathrm{E}_2: \frac{x^2}{\mathrm{~A}^2}+\frac{y^2}{\mathrm{~B}^2}=1,(\mathrm{~A}<\mathrm{B})$ have eccentricity $\frac{4}{5}$. Let the lengths of the latus recta of $\mathrm{E}_1$ and $\mathrm{E}_2$ be $l_1$ and $l_2$, respectively, such that $2 l_1^2=9 l_2$. If the distance between the foci of $E_1$ is 8 , then the distance between the foci of $E_2$ is

If the function $f(x)=\frac{e^x\left(e^{\tan x-x}-1\right)+\log _e(\sec x+\tan x)-x}{\tan x-x}$ is continuous at $x=0$, then the value of $f(0)$ is equal to

Let $(2 \alpha, \alpha)$ be the largest interval in which the function $f(t)=\frac{|t+1|}{t^2}, t<0$, is strictly decreasing. Then the local maximum value of the function $g(x)=2 \log _{\mathrm{e}}(x-2)+\alpha x^2+4 x-\alpha, x>2$, is $\_\_\_\_$

Let a line L passing through the point $\mathrm{P}(1,1,1)$ be perpendicular to the lines $\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1}$ and $\frac{x-17}{1}=\frac{y-71}{1}=\frac{z}{0}$. Let the line L intersect the $y z-$ plane at the point Q . Another line parallel to L and passing through the point $\mathrm{S}(1,0,-1)$ intersects the $y z$-plane at the point R . Then the square of the area of the parallelogram PQRS is equal to $\_\_\_\_$ .