I need to find the coordinates of two vertices with focal points of $(2, 6)$ and $(8, -2)$ and the distance between the vertices is $18$.
I was able to calculate the center of the ellipse which is the midpoint of the foci: $(5, 2)$. I also know that that the $a$ value (the distance from one of the vertices on the major axis to the center) is going to be $9$ since the $c$ value is $5$. I can therefore say that whatever the coordinates of the vertices are must be $4$ units away from the two focal points. However, I am not able to get any further than that in finding the coordinates of the vertices. Any help will be greatly appreciated.
Since you know that the points are colinear and you know their distances from the midpoint, a simple way to find the vertices is to compute the vectors from the midpoint to the foci and scale them to have the right length. You’ve got the midpoint $(5,2)$, so the two vectors are $(2,6)-(5,2)=(-3,4)$ and its negative. The $c$ value is $5$, so these vectors have length $5$ (you can check that for yourself). You need vectors of length $9$: scale them appropriately and add them to the midpoint to get the two vertices.