Find an equation in standard form for the ellipse that satisfies the given condition with major axis endpoints (1,-4) and (1,8), minor axis length 8.

To find the equation in standard form for the ellipse, we can use the general equation for an ellipse centered at the origin:

(x^2 / a^2) + (y^2 / b^2) = 1

where "a" represents the semi-major axis and "b" represents the semi-minor axis.

Since the endpoints of the major axis are given as (1, -4) and (1, 8), we can see that the major axis is vertical and its length is 8 + 4 = 12. Therefore, the semi-major axis (a) will be half of that, which is 12 / 2 = 6.

The minor axis length is given as 8, which means the semi-minor axis (b) will be half of that, which is 8 / 2 = 4.

So now, we can substitute the values of a and b into the equation and rewrite it as:

(x^2 / 6^2) + (y^2 / 4^2) = 1

Simplifying this equation further gives:

(x^2 / 36) + (y^2 / 16) = 1

Therefore, the equation in standard form for the ellipse satisfying the given conditions is:

36x^2 + 16y^2 = 576

The major axis has length 12, and the center is midway between the two points given, or (1,2)

(x-1)^2/16 + (y-2)^2/36 = 1