To find the equation of a hyperbola, we need to understand the key parameters from the given details: the foci and the vertices.
1. Foci: [tex]\((-4, 0)\)[/tex] and [tex]\((4, 0)\)[/tex]
- The distance between the foci is [tex]\(8\)[/tex] units. Since the distance between each focus and the center is [tex]\(2c\)[/tex], we have:
[tex]\[
2c = 8 \implies c = 4
\][/tex]
2. Vertices: [tex]\((-3, 0)\)[/tex] and [tex]\((3, 0)\)[/tex]
- The distance between the vertices is [tex]\(6\)[/tex] units. Since the distance between each vertex and the center is [tex]\(2a\)[/tex], we have:
[tex]\[
2a = 6 \implies a = 3
\][/tex]
3. Relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for a hyperbola:
- For a hyperbola centered at the origin with a horizontal transverse axis, the relationship is:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
4. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 3^2 = 9
\][/tex]
5. Calculate [tex]\(c^2\)[/tex]:
[tex]\[
c^2 = 4^2 = 16
\][/tex]
6. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = c^2 - a^2 = 16 - 9 = 7
\][/tex]
Now, using [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in the standard form of the hyperbola equation:
[tex]\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\][/tex]
Substitute [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex] into the equation:
[tex]\[
\frac{x^2}{9} - \frac{y^2}{7} = 1
\][/tex]
Hence, the equation of the hyperbola centered at the origin with the given foci and vertices is:
[tex]\[
\boxed{\frac{x^2}{9} - \frac{y^2}{7} = 1}
\][/tex]
