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:

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

### Step 1: Simplify the equation
First, we can simplify the equation by dividing everything by \( 25 \) (since \( 5^2 = 25 \)):

\[ \frac{(5x - 10)^2}{25} + \frac{(5y + 15)^2}{25} = \frac{(12x - 5y + 1)^2}{25} \]

This simplifies to:

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

### Step 2: Rearranging the equation
Next, we can express the equation in a more recognizable form. We multiply both sides by \( 25 \):

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

### Step 3: Identify the form of the hyperbola
The left side represents the sum of squares, while the right side is a square of a linear expression. This is indicative of a conic section, specifically a hyperbola.

### Step 4: Finding the eccentricity
The standard form of a hyperbola is given by the relationship between distances from a point to the focus and the directrix. The eccentricity \( e \) can be determined from the coefficients of the conic section.

From the equation derived, we can see that the distance from a point \( P \) to the focus \( S \) is proportional to the distance from \( P \) to the directrix \( D \).

Using the relationship \( Ps = e \cdot Pm \), we can identify that:

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

### Step 5: Calculate \( \frac{25e}{13} \)
Now, we need to find \( \frac{25e}{13} \):

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

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

\[ \boxed{5} \]