We are given a focus at (0,6), the center at the origin, and a vertex at (0,-10), and we are asked for the standard form of the equation for the ellipse.
Let a be the distance from the center to a vertex on the major axis, b the distance from the center to a vertex on the minor axis, and c the distance from the center to one of the foci.
We know the major axis lies on the y-axis as the center is the origin and the foci lie on the y-axis.
Thus a=10. (The length of the major axis is 2a=20; the distance between the vertices on the major axis. By symmetry, the other vertex is at (0,10).)
Also c=6.
The relation between a,b, and c for an ellipse is `b^2=a^2-b^2, a>b ` so
`b^2=100-36=64 ==> b=8 `
Then the equation can be written as `x^2/64+y^2/100=1 ` or `y^2/100+x^2/64=1 ` depending on textbook/instructor preference.
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