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To solve this problem, let's break it down into steps and use the properties of hyperbolas.

### Given Information:
The hyperbola is defined by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
with foci \( F_1 \) and \( F_2 \).

### Focus Points:
For a hyperbola, the foci are located at:
\[ F_1 = (-c, 0) \]
\[ F_2 = (c, 0) \]
where \( c = \sqrt{a^2 + b^2} \).

### Line Passing through \( F_2 \) Parallel to the y-axis:
This line intersects the hyperbola at points \( A \) and \( B \).

### Given Distances:
\[ |F_1A| = 13 \]
\[ |AB| = 10 \]

We need to find the distance of point \( C \) from the center.

### Steps to Solve:

1. **Determine the foci coordinates \( F_1 \) and \( F_2 \):**
   \[ c = \sqrt{a^2 + b^2} \]

2. **Find the coordinates of \( A \) and \( B \):**
   Since the line passing through \( F_2 \) is parallel to the y-axis, we can write the equations as:
   \[ x = c \]

   Substitute \( x = c \) into the hyperbola equation:
   \[ \frac{c^2}{a^2} - \frac{y^2}{b^2} = 1 \]
   \[ \frac{y^2}{b^2} = \frac{c^2}{a^2} - 1 \]
   \[ y^2 = b^2 \left( \frac{c^2 - a^2}{a^2} \right) \]
   \[ y^2 = b^2 \left( \frac{b^2}{a^2} \right) \]
   \[ y^2 = \frac{b^4}{a^2} \]
   \[ y = \pm \frac{b^2}{a} \]

   So the coordinates of \( A \) and \( B \) are:
   \[ A = (c, \frac{b^2}{a}) \]
   \[ B = (c, -\frac{b^2}{a}) \]

3. **Distance Calculation:**
   Given \( |F_1A| = 13 \):
   \[ |F_1A| = \sqrt{(c + c)^2 + \left(\frac{b^2}{a}\right)^2} = 13 \]
   \[ |F_1A| = \sqrt{4c^2 + \frac{b^4}{a^2}} = 13 \]
   \[ 4c^2 + \frac{b^4}{a^2} = 169 \]

4. **Finding \( AB \):**
   Given \( |AB| = 10 \):
   \[ |AB| = \left|\frac{2b^2}{a}\right| = 10 \]
   \[ \frac{2b^2}{a} = 10 \]
   \[ b^2 = 5a \]

5. **Substitute \( b^2 \) into the first equation:**
   \[ b^4 = (5a)^2 = 25a^2 \]
   \[ 4c^2 + \frac{25a^2}{a^2} = 169 \]
   \[ 4c^2 + 25 = 169 \]
   \[ 4c^2 = 144 \]
   \[ c^2 = 36 \]
   \[ c = 6 \]

### Finding the distance of \( C \) from the center:
Given that \( c = 6 \) and \( C \) is located at \( (c, 0) \), the distance of \( C \) from the center (origin) is:
\[ |OC| = 6 \]

So, the distance of point \( C \) from the center is:
\[ \boxed{6} \]

Do you want any further details or have any other questions?

Here are 5 questions you might ask:
1. How do we derive the equations of a hyperbola's asymptotes?
2. Can you explain how to find the eccentricity of a hyperbola?
3. How do the distances between the foci and vertices relate to the hyperbola's properties?
4. How can we determine the equations of lines tangent to a hyperbola?
5. How do hyperbolas differ from other conic sections in terms of their geometric properties?

Tip: When dealing with hyperbolas and their properties, remember to leverage the relationship between the semi-major axis, semi-minor axis, and the distance to the foci, which provides a clear geometric framework for solving related problems.

Explore the solution to a hyperbola problem involving foci, lines parallel to the y-axis, and distance calculations. The problem includes detailed steps to find the distance of point C from the center, integrating key formulas and theorems of conic sections. Suitable for high school students in Grades 10-12.

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