Choose the equation that best represents an ellipse for the given foci and co-vertices. foci (+or- 3,0) co-vertices (o, + or- 6)
describe the number tha make the statement true: 3-16>23
a^2 = b^2 + c^2 on the major axis
a^2 = 6^2 + 3^2
a^2 = 45
x^2/45 + y^2/36 = 1
To determine the equation that represents an ellipse given the foci and co-vertices, we can use the standard form equation for an ellipse, which is:
[(x-h)^2 / a^2] + [(y-k)^2 / b^2] = 1
where (h,k) are the coordinates of the center of the ellipse, and 'a' and 'b' are the semi-major and semi-minor axes, respectively.
First, let's find the center of the ellipse. The coordinates of the center are the average of the foci, which in this case are (+/- 3, 0). Adding the x-coordinates and dividing by 2 gives us (3 + (-3))/2 = 0, and the y-coordinate remains 0. Therefore, the center of the ellipse is (0, 0).
Next, let's find the values of 'a' and 'b', which represent the distance from the center of the ellipse to the vertices and co-vertices.
The distance from the center (0, 0) to the co-vertices (0, +/- 6) is 6. Thus, 'b' equals 6.
The distance from the center (0, 0) to the foci (+/- 3, 0) is also 3. To find 'a', we can use the relationship between 'a', 'b', and the distance between the center and the foci, given by the formula a^2 = b^2 + c^2, where 'c' is the distance between the center and the foci. Plugging in the values, we get:
a^2 = 6^2 + 3^2
a^2 = 36 + 9
a^2 = 45
Taking the square root of both sides, we find that a ≈ √45 ≈ 6.708.
Therefore, the equation that represents the ellipse is:
[(x-0)^2 / 6.708^2] + [(y-0)^2 / 6^2] = 1
Simplifying this equation, we get:
[x^2 / 44.97] + [y^2 / 36] = 1
So, the equation that best represents the ellipse for the given foci (+/- 3,0) and co-vertices (0, +/- 6) is:
[x^2 / 44.97] + [y^2 / 36] = 1