Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:

$25x^2+16y^2=1600$

Find the equation of the ellipse in the following cases:

eccentricity $e=\frac{2}{3}$ and length of latus-rectum $=5$

Find the equation of the ellipse in the following cases:

eccentricity $e=\frac{1}{2}$ and semi-major axis $=4$

Find the equation of the ellipse in the following cases:

eccentricity $e=\frac{1}{2}$ and major axis $=12$

If the equation of the ellipse when the ellipse passes through $(1, 4)$ and $(-6, 1)$ is $m{x}^2+n{y}^2=115$, find the value of $m+n$.

Find the equation of the ellipse in the following cases:

Vertices $(\pm 5,0),$ foci $(\pm 4,0)$