Statement-1: The centre of the circle passing through the points (0, 0), (1, 0) and touching the circle `C : x^(2)+y^(2)=9` lies inside the circle.
Statement-2: If a circle `C_(1)` passes through the centre of the circle `C_(2)` and also touches the circle, the radius of the circle `C_(2)` is twice the radius of circle `C_(1)`

Options

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

Explanation

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

### Step 1: Analyze Statement 1
We need to determine if the center of the circle that passes through the points (0, 0) and (1, 0) and touches the circle defined by the equation \( C: x^2 + y^2 = 9 \) lies inside the circle.

1. **Identify the given circle**: The equation \( x^2 + y^2 = 9 \) represents a circle centered at (0, 0) with a radius of 3.

2. **Identify the points**: The points (0, 0) and (1, 0) are given. The point (0, 0) is the center of the circle \( C \), and (1, 0) lies on the x-axis.

3. **Determine the distance from the center to (1, 0)**: The distance from the center (0, 0) to the point (1, 0) is 1.

4. **Determine the radius of the new circle**: Since the new circle passes through (0, 0) and (1, 0), its center must be on the perpendicular bisector of the segment joining (0, 0) and (1, 0). The midpoint of this segment is (0.5, 0), and the radius of the new circle must be the distance from this center to (0, 0), which is 0.5.

5. **Check if the center lies inside the circle**: The center of the new circle must also touch the circle \( C \). The distance from the center of circle \( C \) (0, 0) to the center of the new circle (0.5, 0) is 0.5, which is less than the radius of circle \( C \) (3). Thus, the center of the new circle lies inside circle \( C \).

**Conclusion for Statement 1**: The statement is true.

### Step 2: Analyze Statement 2
We need to determine if a circle \( C_1 \) that passes through the center of another circle \( C_2 \) and touches it implies that the radius of circle \( C_2 \) is twice the radius of circle \( C_1 \).

1. **Let \( r_1 \) be the radius of circle \( C_1 \)** and \( r_2 \) be the radius of circle \( C_2 \).

2. **Position of the circles**: If circle \( C_1 \) passes through the center of circle \( C_2 \) and touches it, the distance from the center of \( C_2 \) to the center of \( C_1 \) must equal the sum of their radii.

3. **Distance relationship**: The distance from the center of \( C_2 \) to the center of \( C_1 \) is equal to \( r_2 \) (the radius of circle \( C_2 \)) because \( C_1 \) touches \( C_2 \) at the point where \( C_1 \) passes through the center of \( C_2 \).

4. **Set up the equation**: The distance from the center of \( C_2 \) to the center of \( C_1 \) is equal to \( r_1 + r_2 \). Since the center of \( C_1 \) is at a distance \( r_2 \) from the center of \( C_2 \), we have:
\[ r_2 = r_1 + r_2 \]
This implies:
\[ r_2 = 2r_1 \]

**Conclusion for Statement 2**: The statement is true.

### Final Conclusion
Both statements are true.