Final answer:
The equation of an ellipse with center (-5, -2), a major axis of length 6, and an endpoint of the minor axis at (-5, 0) is (x+5)²/9 + (y+2)²/4 = 1, where the semi-major axis is 3 and the semi-minor axis is 2.
Step-by-step explanation:
To find the equation of an ellipse with a center at (-5, -2), a major axis of length 6, and an endpoint of the minor axis at (-5, 0), we first need to determine the lengths of the semi-major and semi-minor axes.
The major axis length is 6, which means the semi-major axis (denoted as a) is half of this, which is 3.
The endpoint of the minor axis at (-5, 0) suggests that the semi-minor axis (denoted as b) extends 2 units from the center (-5, -2) along the y-axis.
Thus, b is 2.
The general equation for an ellipse centered at (h, k) is (x-h)²/a² + (y-k)²/b² = 1. In this case, h is -5 and k is -2.
Plugging the values we have into the equation gives us (x+5)²/3² + (y+2)²/2² = 1, which simplifies to (x+5)²/9 + (y+2)²/4 = 1.
This is the equation of the ellipse.