Let's determine the correct equation for an ellipse centered at the origin, with one vertex at [tex]\((0, 17)\)[/tex] and one focus at [tex]\((0, -8)\)[/tex].
1. Identifying the major axis:
Since the vertex [tex]\((0, 17)\)[/tex] lies on the y-axis, we know this is a vertically oriented ellipse. In such an ellipse, the equation takes the form:
[tex]\[
\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1
\][/tex]
Here, [tex]\(a\)[/tex] is the distance from the center to a vertex, and [tex]\(b\)[/tex] is the distance from the center to a co-vertex.
2. Determining [tex]\(a\)[/tex], the semi-major axis length:
For the given vertex at [tex]\((0, 17)\)[/tex], [tex]\(a = 17\)[/tex].
3. Determining [tex]\(c\)[/tex], the focal distance:
The focus is at [tex]\((0, -8)\)[/tex], so [tex]\(c = 8\)[/tex].
4. Relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
For an ellipse, the relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is given by:
[tex]\[
a^2 = b^2 + c^2
\][/tex]
Using the given values, we get:
[tex]\[
17^2 = b^2 + 8^2
\][/tex]
[tex]\[
289 = b^2 + 64
\][/tex]
[tex]\[
b^2 = 289 - 64
\][/tex]
[tex]\[
b^2 = 225
\][/tex]
[tex]\[
b = \sqrt{225}
\][/tex]
[tex]\[
b = 15
\][/tex]
5. Forming the equation of the ellipse:
Incorporating the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the standard form of the ellipse's equation, we get:
[tex]\[
\frac{y^2}{17^2} + \frac{x^2}{15^2} = 1
\][/tex]
Thus, the correct equation representing the given ellipse is:
[tex]\[
\frac{y^2}{17^2} + \frac{x^2}{15^2} = 1
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{y^2}{17^2}+\frac{x^2}{15^2}=1}
\][/tex]
