To determine the equation of a hyperbola centered at the origin given its foci and vertices, follow these steps:
1. Identify the Values of [tex]\( a \)[/tex] and [tex]\( c \)[/tex]:
- The vertices are given at [tex]\((0, -7)\)[/tex] and [tex]\((0, 7)\)[/tex].
Since the vertices are symmetrically placed about the center (0,0), the distance from the center to a vertex is represented as [tex]\( a \)[/tex]. Therefore:
[tex]\[
a = 7
\][/tex]
- The foci are given at [tex]\((0, -9)\)[/tex] and [tex]\((0, 9)\)[/tex].
Since the foci are symmetrically placed about the center (0,0), the distance from the center to a focus is represented as [tex]\( c \)[/tex]. Therefore:
[tex]\[
c = 9
\][/tex]
2. Use the Relationship Between [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
The relationship for a hyperbola is given by the equation:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
Substituting the known values of [tex]\( a \)[/tex] and [tex]\( c \)[/tex]:
[tex]\[
9^2 = 7^2 + b^2
\][/tex]
Simplifying:
[tex]\[
81 = 49 + b^2
\][/tex]
Solving for [tex]\( b^2 \)[/tex]:
[tex]\[
b^2 = 81 - 49
\][/tex]
[tex]\[
b^2 = 32
\][/tex]
3. Form the Equation of the Hyperbola:
Since the hyperbola is vertical (as the vertices and foci are along the y-axis), the standard form of the equation of a vertical hyperbola centered at the origin is:
[tex]\[
\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
\][/tex]
Substitute [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex]:
[tex]\[
a^2 = 7^2 = 49
\][/tex]
[tex]\[
b^2 = 32
\][/tex]
Thus, the equation of the hyperbola becomes:
[tex]\[
\frac{y^2}{49} - \frac{x^2}{32} = 1
\][/tex]
Therefore, the equation of the hyperbola centered at the origin with the given foci and vertices is:
[tex]\[
\boxed{\frac{y^2}{49} - \frac{x^2}{32} = 1}
\][/tex]
