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The problem depicted in the image involves finding the equation of a hyperbola related to long-range navigation (LORAN) systems, with the foci of the hyperbola marked as \( F(-10, 0) \) and \( F(10, 0) \), and the vertex at \( V(-8, 0) \). This hyperbola's shape suggests a horizontal orientation along the x-axis. The objective is to find the values of \( a \), \( b \), and \( c \) to formulate the hyperbola's equation.

### Key Information:
- **Foci:** \( F_1(-10, 0) \), \( F_2(10, 0) \)
- **Vertex:** \( V(-8, 0) \)
  
### Steps to Solve:

1. **Determine the center of the hyperbola:**
   The center of the hyperbola is located at the midpoint between the two foci. Using the coordinates of the foci \( F_1(-10, 0) \) and \( F_2(10, 0) \), the center \( (h, k) \) is:
   \[
   h = \frac{-10 + 10}{2} = 0, \quad k = 0
   \]
   So, the center of the hyperbola is at \( (0, 0) \).

2. **Find the distance between the center and the vertices (a):**
   The vertex \( V(-8, 0) \) is 8 units away from the center at \( (0, 0) \), so:
   \[
   a = 8
   \]

3. **Find the distance between the center and the foci (c):**
   The foci are located at \( F_1(-10, 0) \) and \( F_2(10, 0) \), and the distance from the center to either focus is:
   \[
   c = 10
   \]

4. **Calculate \( b \) using the relationship \( c^2 = a^2 + b^2 \):**
   \[
   c^2 = a^2 + b^2
   \]
   Substituting the known values of \( c \) and \( a \):
   \[
   10^2 = 8^2 + b^2
   \]
   \[
   100 = 64 + b^2
   \]
   \[
   b^2 = 36 \quad \Rightarrow \quad b = 6
   \]

5. **Write the equation of the hyperbola:**
   Since this is a horizontal hyperbola, the general form of the equation is:
   \[
   \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
   \]
   Substituting \( a^2 = 64 \) and \( b^2 = 36 \):
   \[
   \frac{x^2}{64} - \frac{y^2}{36} = 1
   \]

### Final Answer:
- \( a = 8 \)
- \( b = 6 \)
- \( c = 10 \)
- **Equation of hyperbola:** \( \frac{x^2}{64} - \frac{y^2}{36} = 1 \)

Would you like further details or clarifications?

Here are five related questions to further explore:
1. How would the equation change if the vertices were along the y-axis instead of the x-axis?
2. What is the significance of the relationship \( c^2 = a^2 + b^2 \) in hyperbolas?
3. How do we graph a hyperbola given its equation?
4. What is the eccentricity of this hyperbola, and how is it calculated?
5. Can you derive the equation of a hyperbola when the center is not at the origin?

**Tip:** In hyperbolas, the distance between the center and the foci (denoted by \( c \)) always represents the major axis's impact on the curve’s shape and defines its eccentricity.

Suppose that two long-range navigation (LORAN) stations are the foci of the hyperbolic signal transmissions. A ship is located at a point along the hyperbola. Find the equation of the hyperbola given the foci F(-10, 0) and F(10, 0), and the vertex V(-8, 0).

Learn how to find the equation of a hyperbola given its foci at F(-10, 0) and F(10, 0) and its vertex at V(-8, 0). This problem involves understanding key concepts such as conic sections and the relationship between the foci, vertices, and equation of a hyperbola.

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