To solve the problem step by step, we will analyze each statement regarding the circles A and B.
### Given:
1. Circle A is defined by the equation:
\[
A: (x - 2)^2 + y^2 = 1
\]
This implies that Circle A has a center at \( (2, 0) \) and a radius of \( 1 \).
### Statement 1: Tangents from the origin to Circle A have a radius of \( \frac{1}{3} \) units.
#### Step 1: Find the length of the tangent from the origin to Circle A.
The formula for the length of the tangent from a point \( (x_1, y_1) \) to a circle with center \( (h, k) \) and radius \( r \) is given by:
\[
L = \sqrt{(x_1 - h)^2 + (y_1 - k)^2 - r^2}
\]
For the origin \( (0, 0) \), the center of Circle A is \( (2, 0) \), and the radius \( r = 1 \):
\[
L = \sqrt{(0 - 2)^2 + (0 - 0)^2 - 1^2} = \sqrt{4 - 1} = \sqrt{3}
\]
#### Step 2: Determine the radius of Circle B.
Let the radius of Circle B be \( r_1 \). Since Circle B touches Circle A and the tangents from the origin to Circle A are also tangents to Circle B, we can set up the following relationship using the similarity of triangles:
\[
\frac{r_1}{1} = \frac{1 - r_1}{2}
\]
Cross-multiplying gives:
\[
2r_1 = 1 - r_1 \implies 3r_1 = 1 \implies r_1 = \frac{1}{3}
\]
Thus, **Statement 1 is true**.
### Statement 2: Circle B touching Circle A has a radius of \( 2 \) units.
#### Step 3: Check if the radius can be \( 2 \).
Using the same similarity of triangles approach:
\[
\frac{r_2}{1} = \frac{3 + r_2}{2}
\]
Cross-multiplying gives:
\[
2r_2 = 3 + r_2 \implies r_2 = 3
\]
Since \( r_2 \) cannot be \( 2 \), **Statement 2 is false**.
### Statement 3: The length of common tangents between Circle A and Circle B is \( \sqrt{3} \).
#### Step 4: Calculate the length of the common tangents.
The distance between the centers of Circle A (at \( (2, 0) \)) and Circle B (with center at \( (2 + r_1 + r_2, 0) = (2 + \frac{1}{3} + 3, 0) = (5 + \frac{1}{3}, 0) \)) is:
\[
d = 5 + \frac{1}{3} - 2 = 3 + \frac{1}{3} = \frac{10}{3}
\]
The length of the common tangents \( L \) between two circles with radii \( r_1 \) and \( r_2 \) is given by:
\[
L = \sqrt{d^2 - (r_1 + r_2)^2}
\]
Substituting \( r_1 = \frac{1}{3} \) and \( r_2 = 3 \):
\[
L = \sqrt{\left(\frac{10}{3}\right)^2 - \left(\frac{1}{3} + 3\right)^2} = \sqrt{\frac{100}{9} - \left(\frac{10}{3}\right)^2} = \sqrt{\frac{100}{9} - \frac{100}{9}} = 0
\]
Since the circles touch externally, the length of the common tangents is not \( \sqrt{3} \). Therefore, **Statement 3 is false**.
### Summary of Statements:
- **Statement 1**: True
- **Statement 2**: False
- **Statement 3**: False
### Final Answer:
The answer is: True, False, False.
