Find the coordinates of the center, the foci, and the vertices, sketch the curve, and determine the eccentricity of the ellipse given by the equation

This is an equation for an ellipse in standard form with center at .

with and . We then know its foci are located and where

Therefore, the foci of this ellipse are and . Further, the vertices are given by (page 505 of Apostol) so they are at the points and . Finally, we can compute the eccentricity

from the book the vertices should be at x=a/e and x=-a/e , in this case it should be at x=10*5/4=10.5 and x=-10*5/4=-12.5

im sorry i confused it with vertical lines equation

This makes sense to me, but the vertirx doesnt seem to be equidistant to the line and focus (ie (-10,0) with x = -12.5 and (-8,0). Could someone explain what’s happening here please?

Nevermind, I got it. The distances shouldn’t be equal, but should have a constant ratio represented by the eccentricity (in this case 4/5, or 2/2.5 in what I was observing)