Write an equation in standard form for the ellipse with center (0,0), a Focus at (0,6), and a vertex at (0, -10).

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.