Final answer:
To find the equation of the ellipse, we need to determine its center, vertices, and the distance between the center and the foci. Given that the vertices are located at (-9, -2) and (11, -2), and the directrix is located at x=13.5, we can find and substitute the values of a, b, and the center into the equation of the ellipse.
Step-by-step explanation:
To find the equation of the ellipse, we need to determine its center, vertices, and the distance between the center and the foci.
Given that the vertices are located at (-9, -2) and (11, -2), and the directrix is located at x=13.5, we can determine that the center of the ellipse is the midpoint between the vertices, which is (-9 + 11)/2 = 1 and -2.
The distance between the center and the foci can be found using the distance formula: d = sqrt((c^2) - (a^2)), where c is the distance between the center and one of the foci, and a is the distance from the center to one of the vertices.
Using the information provided, we can find the value of a, which is the distance from the center to one of the vertices: a = 11 - 1 = 10.
Next, we can substitute the values of c and a into the equation to find the value of b in the equation of the ellipse: b = sqrt((c^2) - (a^2)) = sqrt((c^2) - (10^2)).
Finally, we can substitute the values of the center, a, and b into the equation of the ellipse: (x - 1)^2 / 10^2 + (y + 2)^2 / b^2 = 1.