Question 454544
Write an equation for each ellipse described below:
The vertices are at (4,-3) and (4,5) and the foci are at (4,-1) and (4,3). 
I have sketched a graph of these points, but I don't know what to do next.
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Standard form of ellipse with vertical major axis: (x-h)^2/b^2+(y-k)/^a^2=1 (a>b), with (h,k) being the (x,y) coordinates of the center.
Standard form of ellipse with horizontal major axis: (x-h)^2/a^2+(y-k)/^b^2=1 (a>b), with (h,k) being the (x,y) coordinates of the center.
The difference between these two forms is that a^2 and b^2 are interchanged
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From the sketch you just made, you should be able to see that this ellipse has a vertical major axis. You also have the coordinates for the vertices and foci, and therefore, could find a & b.
The center is located midway between the vertices on the major axis:
center (4,1)
length of major axis=8=2a
a=4
a^2=16
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c=length of foci=distance from center to a focus=2
c^2=a^2-b^2
b^2=a^2-c^2=16-4=12
b=√12=3.46..
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Equation of ellipse:
(x-4)^2/12+(y-1)^2/16=1
see the graph below as a visual check on the answer above.

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y=(16-16(x-4)^2/12)^.5+1
{{{ graph( 300, 300, -10, 10, -10, 10,(16-16(x-4)^2/12)^.5+1,-(16-16(x-4)^2/12)^.5+1) }}}