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.