To determine which equation represents a hyperbola with a center at [tex]\((0,0)\)[/tex], a vertex at [tex]\((0,60)\)[/tex], and a focus at [tex]\((0,-65)\)[/tex], we need to understand the properties of hyperbolas.
1. Vertices and Foci:
- For a hyperbola centered at [tex]\((0,0)\)[/tex] and opening along the [tex]\(y\)[/tex]-axis (up and down), the standard form of the equation is:
[tex]\[
\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
\][/tex]
Here, [tex]\(a\)[/tex] is the distance from the center to each vertex, and [tex]\(c\)[/tex] is the distance from the center to each focus.
2. Given Information:
- Vertex at [tex]\((0,60)\)[/tex]: This implies [tex]\(a = 60\)[/tex].
- Focus at [tex]\((0,-65)\)[/tex]: Since it’s symmetric, the focus could also be at [tex]\((0,65)\)[/tex], implying [tex]\(c = 65\)[/tex].
3. Relationship Between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- For hyperbolas, the relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is given by:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
- Substituting the given values:
[tex]\[
65^2 = 60^2 + b^2
\][/tex]
- Solving for [tex]\(b^2\)[/tex]:
[tex]\[
65^2 - 60^2 = b^2
\][/tex]
[tex]\[
4225 - 3600 = b^2
\][/tex]
[tex]\[
b^2 = 625
\][/tex]
4. Equation of the Hyperbola:
- With [tex]\(a = 60\)[/tex] and [tex]\(b = 625^{1/2} = 25\)[/tex], the equation becomes:
[tex]\[
\frac{y^2}{60^2} - \frac{x^2}{25^2} = 1
\][/tex]
Comparing this with the provided options, the correct one is:
[tex]\[
\boxed{\frac{y^2}{60^2} - \frac{x^2}{25^2} = 1}
\][/tex]
