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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.

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  • $\begingroup$ The foci and vertices are colinear. $\endgroup$ – amd Apr 12 '18 at 23:25
  • $\begingroup$ I understand that, but how do I use the fact that they are colinear to solve for the coordinates? Do I write the equation for the line? If so, how do I proceed from there? $\endgroup$ – geo_freak Apr 12 '18 at 23:27
  • $\begingroup$ Writing the equation of the line is fine. Then you know that the vertices are on the line, and you know their distance from the center, so you can solve for them. $\endgroup$ – saulspatz Apr 12 '18 at 23:35
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    $\begingroup$ That works. Or, since you have the coordinates of the center, compute the vectors from the center to the foci and and scale them so that their length is equal to the distance to the vertices. $\endgroup$ – amd Apr 12 '18 at 23:47
  • $\begingroup$ @geo_freak, Use math.stackexchange.com/questions/336622/… $\endgroup$ – lab bhattacharjee Apr 13 '18 at 7:24
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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.

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