What is the general equation of the conic section centered at the origin with a focus at (0, 8) and a sum of distances equal to 20?

To determine the correct general equation of the conic section given the focus at (0, 8) and a sum of distances equal to 20, we first need to identify the type of conic section we are dealing with. The presence of a focus indicates that this is likely an ellipse, especially since the sum of distances from any point on the ellipse to the two foci is a constant value.

In this case, we have one focus at (0, 8). For ellipses, the distance between the foci will define the semi-major axis and semi-minor axis. In the case of a conic section center at the origin with one focus on the y-axis, we can say that the ellipse is vertically oriented.

The semi-major axis (a) can be derived from the given sum of distances. Since the sum of the distances is 20, this means each focal distance contributes to this total. The distance from the center to the focus (c) is 8, and in ellipses, the relationship between the semi-major axis (a), semi-minor axis (b), and the focal distance (c) is given by:

( c = \sqrt{a^2 - b^2}