Final answer:
The equation for the ellipse, based on the given major axis and minor axis length, should be (x + 2)²/16 + (y - 7)²/25 = 1. The center of the ellipse is (-2, 7), with semi-major axis 'a' equal to 5 and semi-minor axis 'b' equal to 4. There is likely an error in the provided options as none match this calculation.
Step-by-step explanation:
To find the equation of an ellipse in standard form that satisfies the given conditions, we need to determine the lengths of the semi-major and semi-minor axes and the coordinates of the center of the ellipse. Since the major axis endpoints are given as (-2, 12) and (-2, 2), we can see that the major axis is vertical and the center of the ellipse will be midway between these endpoints, which is at (-2, (12+2)/2) = (-2, 7). The distance between the major axis endpoints is 12 - 2 = 10, so the length of the semi-major axis is half of that, or 5. This would be our 'a' value. The length of the minor axis is given as 8, so the length of the semi-minor axis is half of that, or 4. This is our 'b' value.
The equation of an ellipse in standard form is (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center of the ellipse. Substituting our values, we get (x + 2)²/4² + (y - 7)²/5² = 1, which simplifies to (x + 2)²/16 + (y - 7)²/25 = 1. However, none of the provided answer choices match this equation. Since the minor axis length is 8, our 'b' value is 4, which would be squared to give b² = 16. The major axis length determined from the endpoints is 10, so a = 5 and a² = 25. Thus, there is likely an error in the question as it appears none of the options provided align with the calculated values from the conditions given.