The equation of the circle passing through (1, 1) and the points of intersection of the circles `x^(2)+y^(2)+13x-3y=0` and `2x^(2)+2y^(2)+4x-7y-25=0` is `4x^(2)+4y^(2)+13x-30y-25=0`.

Answer

To solve the problem, we need to find the equation of the circle that passes through the point (1, 1) and the points of intersection of the circles given by the equations:

1. \( S_1: x^2 + y^2 + 13x - 3y = 0 \)
2. \( S_2: 2x^2 + 2y^2 + 4x - 7y - 25 = 0 \)

We will use the method of combining the two circle equations with a parameter \( \lambda \).

### Step 1: Write the combined equation of the circles
The combined equation of the two circles can be expressed as:
\[ S = S_1 + \lambda S_2 = 0 \]
Substituting the equations of the circles:
\[ x^2 + y^2 + 13x - 3y + \lambda(2x^2 + 2y^2 + 4x - 7y - 25) = 0 \]

### Step 2: Substitute the point (1, 1)
We need to substitute the point (1, 1) into the combined equation to find the value of \( \lambda \):
\[ 1^2 + 1^2 + 13(1) - 3(1) + \lambda(2(1^2) + 2(1^2) + 4(1) - 7(1) - 25) = 0 \]
This simplifies to:
\[ 1 + 1 + 13 - 3 + \lambda(2 + 2 + 4 - 7 - 25) = 0 \]
\[ 12 + \lambda(-24) = 0 \]
\[ \lambda(-24) = -12 \]
\[ \lambda = \frac{1}{2} \]

### Step 3: Substitute \( \lambda \) back into the combined equation
Now substitute \( \lambda = \frac{1}{2} \) back into the combined equation:
\[ S = x^2 + y^2 + 13x - 3y + \frac{1}{2}(2x^2 + 2y^2 + 4x - 7y - 25) = 0 \]
This simplifies to:
\[ x^2 + y^2 + 13x - 3y + (x^2 + y^2 + 2x - \frac{7}{2}y - \frac{25}{2}) = 0 \]
Combining like terms:
\[ (2x^2 + 2y^2) + (13x + 2x) + (-3y - \frac{7}{2}y) - \frac{25}{2} = 0 \]
\[ 2x^2 + 2y^2 + 15x - \frac{13}{2}y - \frac{25}{2} = 0 \]

### Step 4: Multiply through by 2 to eliminate fractions
To eliminate the fractions, multiply the entire equation by 2:
\[ 4x^2 + 4y^2 + 30x - 13y - 25 = 0 \]

### Step 5: Rearranging the equation
Now we will rearrange the equation to match the required form:
\[ 4x^2 + 4y^2 + 30x - 30y - 25 = 0 \]

### Conclusion
Thus, the equation of the circle passing through the point (1, 1) and the points of intersection of the two given circles is:
\[ 4x^2 + 4y^2 + 30x - 30y - 25 = 0 \]