Question

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

Answer 1

The standard equation of the ellipse with the given parameters is (((x - 14)² / 169) + (y² / 144) = 1.

Step-by-step explanation:

The standard equation of an ellipse with vertices at (1, 0) and (27, 0) has a major axis length of 26 units, which gives us the distance from the center to a vertex (the semi-major axis, a) as 13 units.

Given that the eccentricity (e) is 5/13, we can find the focal distance (c) using the formula c = ae, which means c = 13 * (5/13) = 5. The foci are equidistant from the center along the x-axis, thus the center of the ellipse must be the midpoint between the vertices, which is at (14, 0).

The length of the minor axis (2b) can be found using the relationship a2 = b2 + c2, which gives us b as 12. The correct equation of the ellipse would be written with these a2 and b2 values in the denominators and must be centered at the point (14, 0), yielding ((x - 14)2 / 169) + (y2 / 144) = 1.