Find the equation of an ellipse with minor axis length 8 and vertices at (-9,3),(7,3)

distance between vertices is 16 , so a = 8 and we are given that b = 4

centre is the midpoint between (-9,3) and (7,3) which is (-1,3)

equation:
(x+1)^2/64 + (y-3)^2/16 = 1

To find the equation of an ellipse, we need the coordinates of its center and the lengths of its major and minor axes.

Given that the vertices of the ellipse are at (-9, 3) and (7, 3), we can find the center of the ellipse by taking the average of the x-coordinates of the vertices. The y-coordinate of the center will be the same as the y-coordinate of the vertices since the minor axis lies on the line y = 3.

Center of the ellipse = ((-9 + 7) / 2, 3) = (-1, 3)

The length of the minor axis is 8, which means the distance from the center to the endpoints of the minor axis is half of the minor axis length, i.e., 8/2 = 4.

Now we have the center (-1, 3) and half-length of minor axis 4. To find the length of the major axis, we can find the distance between the center and either of the vertices.

Distance between the center (-1, 3) and (-9, 3) = sqrt((-1 - (-9))^2 + (3 - 3)^2) = sqrt(8^2 + 0^2) = sqrt(64) = 8

The major axis length is 8.

Therefore, the equation of the ellipse is:

(x - h)^2/a^2 + (y - k)^2/b^2 = 1, where (h,k) is the center, a is the half-length of the major axis, and b is the half-length of the minor axis.

Plugging in the values, we get:

(x - (-1))^2/8^2 + (y - 3)^2/4^2 = 1

Simplifying, we have:

(x + 1)^2/64 + (y - 3)^2/16 = 1

So, the equation of the ellipse is (x + 1)^2/64 + (y - 3)^2/16 = 1.

To find the equation of an ellipse, we need to determine its center, major axis length, and eccentricity.

Given that the ellipse has minor axis length 8, we can conclude that the distance between the vertices along the x-axis is 16 (from -9 to 7). Therefore, the major axis length is 16.

The center of the ellipse lies at the midpoint of the line segment connecting the two vertices. To find the x-coordinate of the center, we average the x-coordinates of the two vertices: (-9 + 7) / 2 = -1. For the y-coordinate, we take the y-coordinate of one of the vertices, which is 3. Thus, the center of the ellipse is (-1, 3).

Next, we need to determine the eccentricity of the ellipse, denoted by "e." The formula for the eccentricity of an ellipse is e = c/a, where c is the distance from the center to either of the foci and a is half the length of the major axis. We can find c using the Pythagorean theorem: c^2 = a^2 - b^2, where b is half the length of the minor axis.

Given b = 4 (half the length of the minor axis), and a = 8 (half the length of the major axis), we have:

c^2 = 8^2 - 4^2
c^2 = 64 - 16
c^2 = 48

Taking the square root of both sides, we have c ≈ √48 ≈ 6.93.

Now that we have the center (-1, 3), the major axis length (16), and the eccentricity (6.93), we can write the equation of the ellipse in general form:

((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1

where (h, k) denotes the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes, respectively.

Plugging in the values, we get:

((x + 1)^2 / 8^2) + ((y - 3)^2 / 4^2) = 1

Therefore, the equation of the ellipse with a minor axis length of 8 and vertices at (-9,3) and (7,3) is:

((x + 1)^2 / 64) + ((y - 3)^2 / 16) = 1