If the length of the latus rectum rectum of the parabola `169{(x-1)^(2)+(y-3)^(2)}=(5x-12y+17)^(2)` is `L` then the value of `13L/4` is _________.

Explanation

To solve the problem, we need to find the length of the latus rectum \( L \) of the given parabola and then calculate \( \frac{13L}{4} \).

### Step-by-Step Solution:

1. **Rewrite the Given Equation**:
The equation of the parabola is given as:
\[ 169(x-1)^2 + (y-3)^2 = (5x - 12y + 17)^2 \]
We can rewrite this as:
\[ (x-1)^2 + (y-3)^2 = \frac{(5x - 12y + 17)^2}{169} \]

2. **Identify the Focus and Directrix**:
From the equation, we can identify the focus of the parabola. The focus is at the point:
\[ (1, 3) \]
The directrix is given by the line:
\[ 5x - 12y + 17 = 0 \]

3. **Calculate the Distance from the Focus to the Directrix**:
The distance \( d \) from the point \( (1, 3) \) to the line \( 5x - 12y + 17 = 0 \) can be calculated using the formula:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
where \( A = 5 \), \( B = -12 \), \( C = 17 \), and \( (x_0, y_0) = (1, 3) \).

Substituting the values:
\[ d = \frac{|5(1) - 12(3) + 17|}{\sqrt{5^2 + (-12)^2}} = \frac{|5 - 36 + 17|}{\sqrt{25 + 144}} = \frac{|14|}{\sqrt{169}} = \frac{14}{13} \]

4. **Relate the Distance to the Parameter \( a \)**:
The distance \( d \) from the focus to the directrix is equal to \( 2a \):
\[ 2a = \frac{14}{13} \implies a = \frac{14}{26} = \frac{7}{13} \]

5. **Calculate the Length of the Latus Rectum \( L \)**:
The length of the latus rectum \( L \) is given by:
\[ L = 4a = 4 \times \frac{7}{13} = \frac{28}{13} \]

6. **Calculate \( \frac{13L}{4} \)**:
Now we need to find \( \frac{13L}{4} \):
\[ \frac{13L}{4} = \frac{13 \times \frac{28}{13}}{4} = \frac{28}{4} = 7 \]

### Final Answer:
The value of \( \frac{13L}{4} \) is \( \boxed{7} \).