SOLUTION: Find the foci of the ellipse whose major axis has endpoints $(0,0)$ and $(13,0)$ and whose minor axis has length 12. Enter your answer as a list of pairs separated by commas.

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Question 1074902: Find the foci of the ellipse whose major axis has endpoints $(0,0)$ and $(13,0)$ and whose minor axis has length 12.
Enter your answer as a list of pairs separated by commas.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The center of that ellipse is the midpoint of the major axis,
half way between (0,0) and (13,0) on line y=0.
The coordinates of that midpoint are the averages of the coordinates of the vertices.
The midpoint is (6.5,0) , and that is the center of the ellipse.
The distance from the center to each vertex is called the semi-major axis and represented as a .
In this case a=6.5 .
Half of the minor axis is called the semi-minor axis and is represented with b .
That is the distance from the center to each co-vertex.
In this case b=12%2F2=6 .
The distance from the center to each focus is represented with c ,
and is related to a and b by a%5E2=b%5E2%2Bc%5E2 .
In this case 6.5%5E2=6%5E2%2Bc%5E2 .
Solving for c :
6.5%5E2-6%5E2=c%5E2
A calculator can do that, but I don't need one
%286.5%2B6%29%29%286.5-6%29=c%5E2
0.5%2A12.5=c%5E2
6.25=c%5E2
c=sqrt%286.25%29
c=2.5 .
That is the distance from center (6.5,0) to each focus.
The foci are on the same line as the vertices, line y=0,
on either side of the center,
so the x-coordinate of one focus is 6.5-2.5=4 ,
and the x-coordinate of the other focus is 6.5%2B2.5=9 .
So, the foci are highlight%28%22%28+4+%2C+0+%29+%2C+%289+%2C+0+%29%22%29 .