Question 835350
How do I find an equation of an ellipse? It gave me a graph, and a couple points. The major axis is vertical, and the vertex is (0,10) and (0,-10). The focus is at (0,8) and (0,-8).
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When trying to find the equation for a given ellipse, you must first determine whether it has a vertical or horizontal major axis.
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If major axis is vertical,
Standard form of equation for the ellipse: 
{{{(x-h)^2/b^2+(y-k)^2/a^2=1}}}, a>b,(h,k)=(x,y) coordinates of center
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If major axis is horizontal,
Standard form of equation for the ellipse: 
{{{(x-h)^2/a^2+(y-k)^2/b^2=1}}}, a>b,(h,k)=(x,y) coordinates of center
(notice the only change is a^2 and b^2 swapping places)
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Now, for your given ellipse problem:
Gleaned from the 2 given points of the vertex,
x-coordinate of center=0
y-coordinate of center=0
so,center(0,0)
a=10(given distance from center to vertices on the vertical major axis)
a^2=100
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c=8(given distance from center to foci on the vertical major axis)
c^2=64
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c^2=a^2-b^2
b^2=a^2-c^2=100-64=36
b=√36=6
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equation: 
{{{(x-0)^2/b^2+(y-0)^2/a^2=1}}}
{{{x^2/36+y^2/100=1}}}