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The problem involves a hyperbola, with two foci \( F_1 \) and \( F_2 \), and point \( A \) being given at a distance of 3 units from \( F_1 \). The task is to find the distance of point \( A \) from \( F_2 \).

### Step-by-step solution:
Hyperbolas follow the geometric property where the difference of distances from any point on the hyperbola to the two foci is constant and equals \( 2a \), where \( a \) is the distance from the center of the hyperbola to a vertex.

- From the diagram, the foci are located at:
  - \( F_1(-5, 0) \)
  - \( F_2(5, 0) \)
  
  The vertices are:
  - \( V_1(-3, 0) \)
  - \( V_2(3, 0) \)

Thus, the distance between the two vertices is \( 2a = 6 \), so \( a = 3 \).

Now, given that point \( A \) is on the hyperbola and is 3 units away from \( F_1 \), let's use the hyperbola property:

\[
|d(A, F_2) - d(A, F_1)| = 2a = 6
\]

Given that \( d(A, F_1) = 3 \), we can substitute:

\[
|d(A, F_2) - 3| = 6
\]

This leads to two cases:
1. \( d(A, F_2) - 3 = 6 \Rightarrow d(A, F_2) = 9 \)
2. \( -(d(A, F_2) - 3) = 6 \Rightarrow d(A, F_2) = -3 \) (which is not possible, since distance cannot be negative)

Thus, the distance from point \( A \) to \( F_2 \) is **9 units**.

Let me know if you'd like more clarification or details!

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### Follow-up Questions:
1. How is the standard equation of a hyperbola derived using its foci and vertices?
2. What are the basic properties of hyperbolas in terms of their asymptotes?
3. Can you calculate the coordinates of point A using the information given?
4. How does the distance formula apply to finding the distance between two points in the coordinate plane?
5. How can you determine the eccentricity of a hyperbola based on its foci and vertices?

### Tip:
For hyperbolas, remember that the difference of distances from any point on the hyperbola to its two foci remains constant. This helps in solving many geometric problems related to hyperbolas.

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

Learn how to calculate the distance from point A to focus F2, given that point A is 3 units away from focus F1 on a hyperbola. This step-by-step guide explains the geometric properties of hyperbolas and the related distance formula.

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