SOLUTION: Find the equation of the ellipse described. Focus at (0, -3) and (0, 3) vertices at (0, 7) and (0, -7)

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Question 854642: Find the equation of the ellipse described. Focus at (0, -3) and (0, 3) vertices at (0, 7) and (0, -7)
Answer by KMST(5328) About Me  (Show Source):
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With vertices at (0, 7) and (0, -7),
we know that the major axis is the vertical segment connecting (0, 7) and (0, -7),
and that the center of the ellipse is located at (0,0),
the midpoint of the major axis.
We also know that the semi-major axis will be
a=7 , the distance from the center to either one of the vertices.
We know that the focal distance is
c=3 , the distance from the center to the known focus.
We can find b , the semi-minor axis by using the known relation
a%5E2=b%5E2%2Bc%5E2
So 7%5E2=b%5E2%2B3%5E2-->49=b%5E2%2B9-->49-9=b%5E2-->b%5E2=40
Since the equation of an ellipse centered at the origin, with a vertical major axis is
x%5E2%2Fb%5E2%2By%5E2%2Fa%5E2=1
Since b%5E2=40 and a%5E2=7%5E2=49 our equation is
highlight%28x%5E2%2F40%2By%5E2%2F49=1%29

a%5E2=b%5E2%2Bc%5E2 is Pythagoras applied to triangle red%28OBC%29.