Final answer:
The endpoints of the major axis are (8, -3) and (-4, -3), and the endpoints of the minor axis are (2, -3+2√3) and (2, -3-2√3).
Step-by-step explanation:
The given equation of the ellipse is (x-2)^2 / 36 + (y+3)^2 / 12 = 1.
To find the endpoints of the major axis and minor axis, we need to determine the lengths of the semi-major axis and semi-minor axis.
Step 1:
Compare the equation to the standard form of an ellipse:
(x-h)^2 / a^2 + (y-k)^2 / b^2 = 1, where (h, k) is the center of the ellipse.
In this case, the center of the ellipse is (2, -3), so we have:
(x-2)^2 / 36 + (y+3)^2 / 12 = 1
Step 2:
Identify the values of a and b from the equation:
a^2 = 36 and b^2 = 12
Step 3:
Calculate the lengths of the semi-major axis and semi-minor axis:
Length of semi-major axis: a = √36 = 6
Length of semi-minor axis: b = √12 = 2√3
Step 4:
The endpoints of the major axis are located on the x-axis, centered at (2, -3). So the endpoints will be (2+6, -3) and (2-6, -3), which simplify to (8, -3) and (-4, -3).
The endpoints of the minor axis are located on the y-axis, centered at (2, -3). So the endpoints will be (2, -3+2√3) and (2, -3-2√3), which simplify to (2, -3+2√3) and (2, -3-2√3).