Final answer:
To find the equation of the ellipse, we first calculate the distance between the given center and focus as well as the distance between the given center and co-vertex. Using these values as the lengths of the semi-major axis (a) and semi-minor axis (b), respectively, we formulate the ellipse's standard equation: (x + 2)^2/15^2 + (y + 4)^2/12^2 = 1.
Step-by-step explanation:
The question asks us to determine the equation of an ellipse given the center, one of the foci, and a co-vertex. The center of the ellipse is (-2, -4), a focus is located at (-2, 5), and a co-vertex is at (10, -4). We can start by calculating the distance from the center to the focus (the value of c in the ellipse equation) using the distance formula. Since both points have the same x-coordinate, the distance is simply the difference in the y-coordinates, which is |5 - (-4)| = 9.
Next, we calculate the distance from the center to the co-vertex (the value of b in the ellipse equation), which is the semi-minor axis length. The distance is |10 - (-2)| = 12. Now we can use the relationship c^2 = a^2 - b^2 to find a^2. We have already found c = 9 and b = 12, so a^2 = c^2 + b^2 = 9^2 + 12^2 = 81 + 144 = 225. Therefore, a = 15.
The standard form of an ellipse with a horizontal major axis is given by:
\[(x - h)^2/a^2 + (y - k)^2/b^2 = 1\]
where (h, k) is the center of the ellipse.
Substituing the values, we get the equation of the ellipse:
\[(x + 2)^2/15^2 + (y + 4)^2/12^2 = 1\]
This is the equation of the given ellipse.