Question

Find the equation of an ellipse which has its minor axis from (0,0) to (0,4), and which has its foci situated at (5,2) and (−5,2). Use the equation to select all the points from below which are on the ellipse. Note that some of the coordinates are approximate.

a. (4.664,−1)
b. (5,1.743)
c. (4.664,1)
d. (−5,2.743)

Answer 1

An ellipse can be defined by its major axis and minor axis. To find the equation of an ellipse, we need to determine the length of the major and minor axes, as well as the coordinates of the center. The equation of the ellipse in this case is x²/25 + (y - 2)²/4 = 1.

Step-by-step explanation:

An ellipse can be defined by its major axis and minor axis. The major axis is the segment passing through the foci and the center of the ellipse, while the minor axis is the segment perpendicular to the major axis passing through the center of the ellipse.

To find the equation of an ellipse, we need to determine the length of the major and minor axes, as well as the coordinates of the center. In this case, the minor axis has a length of 4, and since it is vertical, the center of the ellipse is located at (0,2).

Next, we can determine the length of the major axis using the distance between the two foci, which is 10 units. Since the foci are symmetric with respect to the center, the coordinates of the foci are (±5,2).

Finally, using the standard form equation of an ellipse, we get:

(x - h)²/a² + (y - k)²/b² = 1

where (h,k) is the center of the ellipse, and a and b are the lengths of the major and minor axes, respectively.

Substituting the values, we have:

(x - 0)²/5² + (y - 2)²/2² = 1

So the equation of the ellipse is x²/25 + (y - 2)²/4 = 1.