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(52798)  (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.
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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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