Statement-1: The equation of a circle through the origin and belonging to the coaxial system, of which limiting points are (1, 1) and (3, 3) is `2x^(2)+2y^(2)-3x-3y=0`
Statement-2: The equation of a circle passing through the points (1, 1) and (3, 3) is `2x^(2)+y^(2)-2x-6y+6=0`.

Options

A. Option1 Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Option2 Statement-1 is True, Statement-2 is not a correct explanation for Statement-1.
C. Option3 Statement-1 is True, Statement-2 is False.
D. Option4 Statement-1 is False, Statement-2 is True.

Explanation

To solve the given problem, we will analyze both statements step by step.

### Statement 1:
We need to determine if the equation of the circle through the origin and belonging to the coaxial system, with limiting points (1, 1) and (3, 3), is given by \(2x^2 + 2y^2 - 3x - 3y = 0\).

1. **Identify the limiting points**: The limiting points are (1, 1) and (3, 3). The center of the coaxial circles can be found by averaging the coordinates of the limiting points.
\[ \text{Center} = \left( \frac{1 + 3}{2}, \frac{1 + 3}{2} \right) = (2, 2) \]

2. **Form the general equation of the coaxial circle**: The equation of the coaxial system can be expressed as:
\[ (x - 1)^2 + (y - 1)^2 + \lambda \left( (x - 3)^2 + (y - 3)^2 \right) = 0 \]
where \(\lambda\) is a parameter.

3. **Substituting the origin (0, 0)**: Since the circle passes through the origin, substitute \(x = 0\) and \(y = 0\):
\[ (0 - 1)^2 + (0 - 1)^2 + \lambda \left( (0 - 3)^2 + (0 - 3)^2 \right) = 0 \]
Simplifying:
\[ 1 + 1 + \lambda(9 + 9) = 0 \implies 2 + 18\lambda = 0 \implies \lambda = -\frac{1}{9} \]

4. **Substituting \(\lambda\) back into the equation**: Now we substitute \(\lambda\) back into the coaxial equation:
\[ (x - 1)^2 + (y - 1)^2 - \frac{1}{9} \left( (x - 3)^2 + (y - 3)^2 \right) = 0 \]
Multiplying through by 9 to eliminate the fraction:
\[ 9((x - 1)^2 + (y - 1)^2) - (x - 3)^2 - (y - 3)^2 = 0 \]

5. **Expanding and simplifying**:
\[ 9(x^2 - 2x + 1 + y^2 - 2y + 1) - (x^2 - 6x + 9 + y^2 - 6y + 9) = 0 \]
\[ 9x^2 + 9y^2 - 18x - 18y + 18 - x^2 + 6x - 9 - y^2 + 6y - 9 = 0 \]
\[ (9x^2 - x^2) + (9y^2 - y^2) + (-18x + 6x) + (-18y + 6y) + (18 - 9 - 9) = 0 \]
\[ 8x^2 + 8y^2 - 12x - 12y = 0 \]
Dividing by 4:
\[ 2x^2 + 2y^2 - 3x - 3y = 0 \]
This confirms that Statement 1 is true.

### Statement 2:
We need to check if the equation \(2x^2 + y^2 - 2x - 6y + 6 = 0\) represents a circle passing through the points (1, 1) and (3, 3).

1. **Substituting the points into the equation**:
- For point (1, 1):
\[ 2(1)^2 + (1)^2 - 2(1) - 6(1) + 6 = 2 + 1 - 2 - 6 + 6 = 1 \quad (\text{not } 0) \]
- For point (3, 3):
\[ 2(3)^2 + (3)^2 - 2(3) - 6(3) + 6 = 2(9) + 9 - 6 - 18 + 6 = 18 + 9 - 6 - 18 + 6 = 9 \quad (\text{not } 0) \]

Since neither point satisfies the equation, Statement 2 is false.

### Conclusion:
- **Statement 1** is true.
- **Statement 2** is false.