Which of the following is the standard equation of the ellipse with vertices at (1, 0) and (27, 0) and an eccentricity of 5/13?
quantity x plus 14 end quantity squared over 144 plus y squared over 169 equals 1
quantity x plus 14 end quantity squared over 25 plus y squared over 169 equals 1
quantity x minus 14 end quantity squared over 169 plus y squared over 144 equals 1
quantity x minus 14 end quantity squared over 169 plus y squared over 25 equals 1
you are very strange. After all that work, you go from
[(x - 14)^2 / 169] + [(y - 0)^2 / 256] = 1
to
quantity x minus 14 end quantity squared over 169 plus y squared over 25 equals 1
That is,
(x - 14)^2 / 169 + y^2/25 = 1
It's not as though a bot makes typos. How did you lose the 6?
when it should be
(x - 14)^2 / 169 + y^2/256 = 1
Apologies for the error. You are correct. The correct equation should be:
[(x - 14)^2 / 169] + [(y - 0)^2 / 256] = 1
which simplifies to:
(x - 14)^2 / 169 + y^2 / 256 = 1
Thank you for catching that mistake.
The standard equation for an ellipse with vertices on the x-axis, center at (h, k), and semi-major axis length a can be written as:
[(x - h)^2 / a^2] + [(y - k)^2 / b^2] = 1
In this case, the vertices are at (1, 0) and (27, 0). The distance between the vertices is equal to 2a, so a = (27 - 1) / 2 = 13.
The eccentricity is given as 5/13, which is the ratio of the distance from the center to one of the foci (c) to the length of the semi-major axis (a). Therefore, c = (5/13) * 13 = 5.
Since the ellipse has vertices on the x-axis, the center is at the midpoint between the vertices: (h, k) = ((1 + 27) / 2, 0) = (14, 0).
Plugging in the values, we have:
[(x - 14)^2 / 13^2] + [(y - 0)^2 / b^2] = 1
To solve for b^2, we can use the relationship between a, b, and c in an ellipse, which is given by:
c^2 = a^2 - b^2
(5/13)^2 = 13^2 - b^2
25/169 = 169 - b^2
b^2 = 169 - 25/169
b^2 = (169^2 - 25) / 169
b^2 = 28144 / 169
b^2 = 16^2
b = 16
Therefore, the standard equation of the given ellipse is:
[(x - 14)^2 / 169] + [(y - 0)^2 / 256] = 1
This matches with the answer choice:
quantity x minus 14 end quantity squared over 169 plus y squared over 25 equals 1