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This diagram shows a hyperbola with foci \( F_1 \) and \( F_2 \), and vertices \( V_1 \) and \( V_2 \). The point \( A \) is 3 units away from the focus \( F_1 \), and the problem asks how far point \( A \) is from \( F_2 \).

### Step-by-step explanation:

1. **Recognizing the Hyperbola Properties**:  
   In a hyperbola, the difference in distances between any point on the hyperbola and the two foci \( F_1 \) and \( F_2 \) is constant. This difference is equal to the distance between the vertices \( V_1 \) and \( V_2 \).

2. **Vertices and Foci Positions**:  
   From the diagram, we can observe:
   - \( F_1 \) is located at \( (-5, 0) \).
   - \( F_2 \) is located at \( (5, 0) \).
   - \( V_1 \) is at \( (-4, 0) \), and \( V_2 \) is at \( (4, 0) \).
   
   The distance between the vertices \( V_1 \) and \( V_2 \) is 8 units, so this is the constant difference in the distances between any point on the hyperbola and the two foci.

3. **Distance Information**:
   Given that point \( A \) is 3 units away from \( F_1 \), we know that the difference between the distances to \( F_1 \) and \( F_2 \) is 8. Let's call the distance from \( A \) to \( F_2 \) as \( x \).

   This gives us the equation:
   \[
   |x - 3| = 8
   \]

4. **Solving for \( x \)**:
   This absolute value equation splits into two cases:
   \[
   x - 3 = 8 \quad \text{or} \quad x - 3 = -8
   \]
   Solving these:
   \[
   x = 11 \quad \text{or} \quad x = -5
   \]

   Since distance cannot be negative, the distance from point \( A \) to \( F_2 \) is **11 units**.

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### Related Questions:
1. What are the general properties of hyperbolas?
2. How do we derive the equation for the constant difference between distances in a hyperbola?
3. How does the location of the foci affect the shape of a hyperbola?
4. How is the eccentricity of a hyperbola calculated?
5. What role do asymptotes play in the geometry of hyperbolas?

### Tip:
In a hyperbola, the constant difference between the distances to the foci can be helpful in many problems, especially when dealing with distance-related questions.

If point A is 3 units away from F1, how far is point A from F2?

In this hyperbola problem, point A is 3 units away from F1, and we calculate how far point A is from F2. This example uses the constant difference in distances property of hyperbolas to solve for the unknown distance.

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