Question

Find the equation of an ellipse which has its major axis from (0,−5) to (0,5), and which has its foci situated at (0,−3) and (0,3). Use the equation to select all the points from below which are on the ellipse. Note that some of the coordinates are approximate.

a. (−2,−4.330)
b. (1,4.841)
c. (−2,4.330)
d. (1,−4.841)

Answer 1

The equation of the ellipse is
`\( (y^2)/(9) + (x^2)/(16) = 1 \)`. Among the given points, only (1, 4.841) lies on the ellipse. Hence, the correct answer is b. (1, 4.841).

The given ellipse has its major axis along the y-axis, and the foci are vertically aligned at (0, -3) and (0, 3). The distance from the center to each focus is the value of 'c' in the ellipse equation.

The formula for the ellipse with a vertical major axis is

`\( (x^2)/(b^2) + (y^2)/(a^2) = 1 \)`, where a is the semi-major axis, and b is the semi-minor axis.

The distance between the foci is 2c, and in this case, it's
`\( 2 * 3 = 6 \)`. So,
2c = 6 implies c = 3 .

Since the ellipse is centered at the origin, the equation becomes

`\( (y^2)/(3^2) + (x^2)/(b^2) = 1 \)`.

Now, we need to find the value of b . The distance between the vertices (endpoints of the major axis) is the value of 2a, which is 10 in this case. Therefore, 2a = 10 implies a = 5 .

Now, b can be found using
`\( a^2 = b^2 - c^2 \)`. Substituting the known values,
`\( 5^2 = b^2 - 3^2 \)`, and solving gives b = 4.

So, the equation of the ellipse is
`\( (y^2)/(9) + (x^2)/(16) = 1 \)`.

Now, checking the given points:

a. (-2, -4.330): Not on the ellipse.

b. (1, 4.841): On the ellipse.

c. (-2, 4.330): Not on the ellipse.

d. (1, -4.841): Not on the ellipse.

Therefore, the correct answer is b- (1, 4.841).