If e is the eccentricity of the hyperbola `(5x - 10)^(2) + (5y + 15)^(2) = (12x - 5y + 1)^(2)` then `(25e)/13` is equal to _____

Answer

To solve the problem, we need to find the eccentricity \( e \) of the hyperbola given by the equation:

\[ (5x - 10)^{2} + (5y + 15)^{2} = (12x - 5y + 1)^{2} \]

and then calculate \( \frac{25e}{13} \).

### Step 1: Rewrite the equation
First, we can simplify the equation by factoring out constants from the left-hand side:

\[ (5(x - 2))^{2} + (5(y + 3))^{2} = (12x - 5y + 1)^{2} \]

This can be rewritten as:

\[ 25((x - 2)^{2} + (y + 3)^{2}) = (12x - 5y + 1)^{2} \]

### Step 2: Identify the form of the hyperbola
We need to express this in a standard form of a hyperbola. The general form of a hyperbola can be expressed as:

\[ \frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1 \]

or

\[ \frac{(y - k)^{2}}{b^{2}} - \frac{(x - h)^{2}}{a^{2}} = 1 \]

### Step 3: Find the distance from the center to the foci
From the equation, we can see that the left-hand side represents a circle scaled by a factor of 25, and the right-hand side represents a quadratic equation. To find the eccentricity \( e \), we need to identify the distances involved.

The distance from the center to the foci \( c \) is given by the formula:

\[ c = \sqrt{a^{2} + b^{2}} \]

where \( e = \frac{c}{a} \).

### Step 4: Calculate the eccentricity
From the analysis, we can derive that the eccentricity \( e \) is given by:

\[ e = \frac{13}{5} \]

### Step 5: Calculate \( \frac{25e}{13} \)
Now we can substitute the value of \( e \) into the expression \( \frac{25e}{13} \):

\[ \frac{25e}{13} = \frac{25 \times \frac{13}{5}}{13} \]

This simplifies to:

\[ \frac{25 \times 13}{5 \times 13} = \frac{25}{5} = 5 \]

### Final Answer
Thus, the value of \( \frac{25e}{13} \) is:

\[ \boxed{5} \]