SOLUTION: An ellipse has vertices (2 − √ 61, 5) and (2 + √ 61, 5), and its minor axis is 12 units long. Find its standard equation and its foci.

Question 1186658: An ellipse has vertices (2 − √ 61, 5) and (2 + √ 61, 5), and its minor axis is 12 units long. Find its standard equation and its foci.

Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

An ellipse has vertices (, ) and (, ), and its minor axis is units long.
Find its standard equation and its foci.
center is half way between vertices
(,)=(,)=(,)=> and

minor axis is =>

........plug in known

...............plug in coordinates of vertices (, )

so, your equation is:

for an ellipse with major axis parallel to the x-axis, the Foci (focus ) are defined as :
(, ), (, )
find

or
(,)=(, )
(,) =(,)

Answer by ikleyn(52800)   (Show Source): You can put this solution on YOUR website!
.
An ellipse has vertices (2 − √ 61, 5) and (2 + √ 61, 5), and its minor axis is 12 units long.
Find its standard equation and its foci.
~~~~~~~~~~~~~~

            It can be done (and it should be done) in much shorter way, than @MathLover1 does it.

            It does not require so intensive calculations.

Looking at the foci coordinates, you see that they are on the horizontal line y = 5. So, the major axis is horizontal, parallel to x-axis, and the length of the horizontal axis is - = . Hence, the length of the major semi-axis "a" is half of it, i.e. a = . The length of the minor semi-axis is b = = 6. The center of the ellipse is the point (2,5). THEREFORE, the standard form equation of the ellipse is + = 1. ANSWER The distance from the center to the focus is c = = = = 5. The foci are (2+5,5) = (7,5) and (2-5,5) = (-3,5).

Solved.

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