To determine which equation represents the correct ellipse, we need to derive the properties of the ellipse from the given information: the major axis length is 18 and the foci are located at [tex]\((4,7)\)[/tex] and [tex]\((4,11)\)[/tex].
1. Calculate the distance between the foci:
The foci are [tex]\((4,7)\)[/tex] and [tex]\((4,11)\)[/tex]. The distance between the foci, denoted as [tex]\(2c\)[/tex], can be determined by the vertical distance between these points:
[tex]\[
2c = |11 - 7| = 4
\][/tex]
Hence, [tex]\(c = \frac{4}{2} = 2\)[/tex].
2. Determine the semi-major axis [tex]\(a\)[/tex]:
The length of the major axis is 18, so the semi-major axis [tex]\(a\)[/tex] is:
[tex]\[
a = \frac{18}{2} = 9
\][/tex]
3. Find the semi-minor axis [tex]\(b\)[/tex]:
Using the relationship between the axes in an ellipse, [tex]\(c^2 = a^2 - b^2\)[/tex]:
[tex]\[
c^2 = a^2 - b^2 \implies 2^2 = 9^2 - b^2 \implies 4 = 81 - b^2 \implies b^2 = 81 - 4 = 77
\][/tex]
Therefore, [tex]\(b = \sqrt{77}\)[/tex].
4. Find the center of the ellipse:
The center of the ellipse is the midpoint of the foci:
[tex]\[
\left( \frac{4+4}{2}, \frac{7+11}{2} \right) = (4, 9)
\][/tex]
5. Formulate the standard equation of the ellipse:
The standard form for a vertically oriented ellipse (because the foci are vertically aligned) centered at [tex]\((h,k)\)[/tex] is:
[tex]\[
\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1
\][/tex]
Substituting [tex]\(h = 4\)[/tex], [tex]\(k = 9\)[/tex], [tex]\(a^2 = 81\)[/tex], and [tex]\(b^2 = 77\)[/tex]:
[tex]\[
\frac{(x-4)^2}{77} + \frac{(y-9)^2}{81} = 1
\][/tex]
Therefore, the correct equation of the ellipse is:
D. [tex]\(\frac{(x-4)^2}{77}+\frac{(y-9)^2}{81}=1\)[/tex]
