Final answer:
The correct equation for the ellipse with given foci and sum of distances is (x + 8)^2/25 + y^2/16 = 1, which corresponds to Option A.
Step-by-step explanation:
An ellipse is a closed curve such that the sum of the distances from any point on the curve to the two foci is a constant. Given the foci of the ellipse at F1(-8,0) and F2(8,0) and that the sum of distances to the foci is 20, we need to determine which equation corresponds to this ellipse. The center of the ellipse in this case will be the midpoint between the foci, which is at the origin (0,0).
Furthermore, the sum of the distances from any point on the ellipse to its foci is equal to the length of the major axis, which in this case is 20. Hence, the major axis is 20 and since the distance between the foci is 16 (from -8 to 8), we can calculate the length of the semi-major axis 'a' and semi-minor axis 'b' using the relationship c^2 = a^2 - b^2, where 2c is the distance between the foci, and 2a is the length of the major axis.
So, a = 20/2 = 10, and c = 16/2 = 8. Now, we can find b, b^2 = a^2 - c^2 = 10^2 - 8^2 = 100 - 64 = 36, thus, b = 6. The standard form for the equation of an ellipse centered at the origin with horizontal major axis is (x^2/a^2) + (y^2/b^2) = 1. Substituting a and b yields the equation (x^2/10^2) + (y^2/6^2) = (x^2/100) + (y^2/36) = 1.
Therefore, the equation that represents the given ellipse is Option A: ((x + 8)^2/25) + (y^2/16) = 1. Here, 25 corresponds to the square of the semi-major axis (a^2), and 16 corresponds to the square of the semi-minor axis (b^2).