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To find the equation of the hyperbola centered at the origin with the given foci and vertices, follow these steps:

1. **Identify the coordinates of the foci and vertices**:
   - Foci: \((-9,0)\) and \((9,0)\)
   - Vertices: \((-2,0)\) and \((2,0)\)

2. **Determine the values of \(c\) and \(a\)**:
   - The distance from the center to each focus is \(c\). Here, \(c = 9\).
   - The distance from the center to each vertex is \(a\). Here, \(a = 2\).

3. **Calculate \(b\)**:
   - Use the relationship \(c^2 = a^2 + b^2\).
   - Substitute the known values: \(9^2 = 2^2 + b^2\)
   - Solve for \(b\):
     \[
     81 = 4 + b^2
     \]
     \[
     b^2 = 77
     \]
     \[
     b = \sqrt{77}
     \]

4. **Write the standard form of the hyperbola's equation**:
   - Since the transverse axis is horizontal, the equation is of the form:
     \[
     \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
     \]
   - Substitute \(a^2\) and \(b^2\) with the values found:
     \[
     \frac{x^2}{4} - \frac{y^2}{77} = 1
     \]

So, the equation of the hyperbola is:
\[
\frac{x^2}{4} - \frac{y^2}{77} = 1
\]

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How do you find the asymptotes of a hyperbola?
2. What is the significance of the transverse and conjugate axes in a hyperbola?
3. How does the orientation of a hyperbola affect its equation?
4. Can you explain the relationship between the distances in a hyperbola (foci, vertices, and co-vertices)?
5. How do you graph a hyperbola given its equation?

**Tip:** Always double-check the distances used for \(a\), \(b\), and \(c\) in hyperbola problems to avoid common calculation errors.

This hyperbola is centered at the origin. Find its equation. Foci: (-9,0) and (9,0) Vertices: (-2,0) and (2,0)

Learn how to find the equation of a hyperbola centered at the origin with foci at (-9,0) and (9,0), and vertices at (-2,0) and (2,0). This detailed step-by-step solution includes key concepts like conic sections and the relationship between a, b, and c in hyperbolas. Suitable for high school students in Grades 10-12.

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