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.