Write an equation of the ellipse with foci at (0, ±10), and vertices at (0, ±11).
AAAaannndd the bot gets it wrong yet again!
b^2 = (11^2 - 10^2) = 21 since the axis is vertical, we have
x^2/21 + y^2/121 = 1
Apologies for the mistake. Thank you for correcting it. The equation of the ellipse with foci at (0, ±10) and vertices at (0, ±11) is:
x^2/21 + y^2/121 = 1
Here are the answers for the rest of the test.
5.7.1 - Test: Quadratic Relations and Conic Sections Unit Test Part 1
1. A
2. C
3. B
4. B
5. C
6. B
7. A
8. A
9. B
10. A
11. D
12. B
13. C
14. A
Thank you for providing the answers. As an AI language model, I don't have any context about the questions, however, I hope these answers are helpful for the person who needs them.
To write the equation of an ellipse, we need to know the coordinates of the foci and the vertices. For an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant.
Given the foci at (0, ±10) and the vertices at (0, ±11), we can determine key information about the ellipse.
The center: The center of the ellipse is at the midpoint between the two foci. Since the y-coordinates of the foci are ±10, the center has a y-coordinate of 0.
The distance from the center to each focus is 10, and the distance from the center to each vertex is 11. Therefore, we can determine the value for a (which is the distance from the center to a vertex) and c (which is the distance from the center to a focus) using the Pythagorean theorem:
a^2 = c^2 + b^2
where a is the major radius, b is the minor radius, and c is the distance from the center to a focus.
Applying this formula, we find:
11^2 = 10^2 + b^2
121 = 100 + b^2
b^2 = 21
Now that we know the values of a and b, we can write the equation of the ellipse:
The equation of an ellipse with the center at (h, k) is:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
In this case, since the center is (0, 0) and we have found values for a^2 (which is 21) and b^2 (which is 21), the equation of the ellipse is:
x^2 / 21 + y^2 / 21 = 1
The center of the ellipse is at (0,0), halfway between the foci. The distance from the center to each focus is c=10. The distance from the center to each vertex is a=11. Therefore, the length of the major axis is 2a=22, and the length of the minor axis is 2b=sqrt(22^2-10^2)=sqrt(384). The equation of the ellipse is:
x^2/a^2 + y^2/b^2 = 1
Substitute a=11 and b=sqrt(384):
x^2/121 + y^2/384 = 1
Simplify:
384x^2 + 121y^2 = 46656
Answer: 384x^2 + 121y^2 = 46656.