Find the standard form of the equation of the ellipse with the given characteristics. foci: endpoints of the major axis:

Question

Find the standard form of the equation of the ellipse with the given characteristics. foci:  endpoints of the major axis:

Quizplus · Accepted Answer

The distance between the foci is given by c, which is 6. The distance between the endpoints of the major axis is given by 2a, which is also 6. Thus, the center of the ellipse is at the midpoint of the major axis, which is (4, -3). 
We know that the distance from the center to each focus is given by c, so we can find the distance to each focus as 6/2 = 3. One focus is at (4, -6) and the other is at (4, 0). 
We also know that the equation of an ellipse centered at (h, k) with semi-major axis a and semi-minor axis b is given by:
(x-h)^2 / a^2 + (y-k)^2 / b^2 = 1
We have the center at (4, -3) and the endpoints of the major axis are (4, 0) and (4, -6). The distance between the center and each endpoint is given by the semi-major axis a, so we can find a as 3. The distance between the center and the minor axis is given by the semi-minor axis b. Since the ellipse is symmetric, we can see that this distance is 3 as well. Thus, the equation of the ellipse is:
(x-4)^2 / 3^2 + (y+3)^2 / 3^2 = 1 
Simplifying this equation gives us D as the correct choice.

Find the Standard Form of the Equation of the Ellipse