Find the equation of the ellipse with the following properties. the ellipse with foci at (0, 6) and (0, -6); y-intercepts (0, 8) and (0, -8).

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The general equation of a horizontal ellipse is

(x-h)2/a2 + (y-k)2/b2 = 1, at center (h,k) while a = semi-major axis, b = semi-minor axis. These are related through the distance of the focus from the center,c. a2 = b2 + c2.

If you draw the points on a coordinate plane, the center of the ellipse is at (0,0), so h and k equals 0. Then, the minor axis (2b) spans from 8 to -8 of the y-axis. This is equal to 16 units. Hence,

2b = 16
b = 8
b^2 = 64

The distance between the two foci is 2c. Thus,

2c = 12
c = 6
c^2 = 36

Then, a2 = 64 + 36 = 100. Substituting to the general equation:

x^2/100 + y^2/64 = 1