Statement 1`:` The number of common tangents to the circles `x^(2) + y^(2) =4 ` and `x^(2) + y^(2) -6x - 6y = 24` is 3.
Statement 2 `:` If two circles touch each other externally thenit has two direct common tangents and one indirect common tangent.

Options

A. Statement-1 is true, Statement-2 is false
B. Statement-1
:
is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement -1
D. Statement-1 is False, Statement-2 is True

Answer

To analyze the given statements about the circles, we will first determine the properties of the circles involved and then verify the statements step by step.

### Step 1: Identify the circles
The first circle is given by the equation:
\[ x^2 + y^2 = 4 \]
This can be rewritten to identify its center and radius:
- Center \( C_1 = (0, 0) \)
- Radius \( r_1 = \sqrt{4} = 2 \)

The second circle is given by the equation:
\[ x^2 + y^2 - 6x - 6y = 24 \]
We can rearrange this into standard form by completing the square:
\[ (x^2 - 6x) + (y^2 - 6y) = 24 \]
Completing the square:
\[ (x - 3)^2 - 9 + (y - 3)^2 - 9 = 24 \]
\[ (x - 3)^2 + (y - 3)^2 = 24 + 18 = 42 \]
Thus, we have:
- Center \( C_2 = (3, 3) \)
- Radius \( r_2 = \sqrt{42} \approx 6.48 \)

### Step 2: Calculate the distance between the centers
The distance \( d \) between the centers \( C_1 \) and \( C_2 \) is calculated using the distance formula:
\[ d = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \approx 4.24 \]

### Step 3: Analyze the relationship between the circles
We need to compare the distance \( d \) with the sum and difference of the radii to determine the number of common tangents.

1. **Sum of the radii**:
\[ r_1 + r_2 = 2 + \sqrt{42} \approx 2 + 6.48 = 8.48 \]

2. **Difference of the radii**:
\[ r_2 - r_1 = \sqrt{42} - 2 \approx 6.48 - 2 = 4.48 \]

### Step 4: Determine the number of common tangents
Now we compare \( d \) with \( r_1 + r_2 \) and \( |r_2 - r_1| \):
- Since \( d \approx 4.24 < 8.48 \) (the sum of the radii), the circles are not separate.
- Since \( d \approx 4.24 < 4.48 \) (the difference of the radii), one circle lies inside the other.

### Conclusion
Since one circle lies inside the other and they do not touch, there are no common tangents between the two circles.

### Verification of Statements
- **Statement 1**: The number of common tangents to the circles is **not 3**. This statement is **false**.
- **Statement 2**: If two circles touch each other externally, they have two direct common tangents and one indirect common tangent. This statement is **true**.

### Final Answer
- Statement 1 is false.
- Statement 2 is true.