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Question 385029: Write the equation of the ellipse that meets each set of conditions.
The foci are at (-1,1) and (-1,5) and at a=7
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Since the  -coordinates of the foci are equal, we know that the major axis is a vertical line, 
The center of the ellipse is the midpoint of the segment that joins the foci so we use the midpoint formulas:
}{2}\ =\ -1)
and
Hence, center is (-1, 3)
The denominators in the ellipse equation are  and  and since the major axis is vertical, the goes with the  term. These two quantities are related by the equation:
Where  is the measure of the segment between either focus and the center.
You can either use the distance formula or, in this case, by inspection to find the value of , namely 2. We are also given that 
So:
so
Now we have enough information to write the desired equation:
^2}{b^2}\ +\ \frac{(y\ -\ k)^2}{a^2}\ =\ 1)
where ) is the center of the ellipse, ) are the foci, the semi-major axis measures  and therefore the vertices are at ) , the semi-minor axis measures  and therefore the endpoints of the semi-minor axis are )
Hence your equation is:
^2}{45}\ +\ \frac{(y\ -\ 3)^2}{49}\ =\ 1)
John
My calculator said it, I believe it, that settles it
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