Line `L_(1)-=3x-4y+1=0` touches the cirlces `C_(1) and C_(2)`. Centres of `C_(1) and C_(2)` are `A_(2)(1, 2) and A_(2)(3, 1)` respectively, then identify the INCORRECT statement from the following statements.

Options

A. `L_(1)` is direct common tangent of these circles
B. `L_(1)` is transverse common tangent to these circles
C. Radius of circle `C_(1)` is `(4)/(5)` units
D. Radius of circle `C_(2)` is `(6)/(5)` units

Answer

To solve the problem step by step, we need to analyze the given line and the circles, and then determine which statement is incorrect.

### Step 1: Identify the given line and circles
The line is given by the equation:
\[ L_1: 3x - 4y + 1 = 0 \]
The centers of the circles are:
- Circle \( C_1 \) with center \( A_1(1, 2) \)
- Circle \( C_2 \) with center \( A_2(3, 1) \)

### Step 2: Determine the type of tangent
The problem states that the line \( L_1 \) touches both circles. We need to determine if it is a direct common tangent or a transverse common tangent.

- **Direct Common Tangent (DCT)**: A tangent that touches both circles on the same side.
- **Transverse Common Tangent (TCT)**: A tangent that touches both circles on opposite sides.

### Step 3: Calculate the distance from the centers to the line
We will use the formula for the distance \( d \) from a point \( (x_1, y_1) \) to a line \( ax + by + c = 0 \):
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]

#### For Circle \( C_1 \) at \( A_1(1, 2) \):
- \( a = 3, b = -4, c = 1 \)
\[ d_1 = \frac{|3(1) - 4(2) + 1|}{\sqrt{3^2 + (-4)^2}} = \frac{|3 - 8 + 1|}{\sqrt{9 + 16}} = \frac{|-4|}{5} = \frac{4}{5} \]

#### For Circle \( C_2 \) at \( A_2(3, 1) \):
\[ d_2 = \frac{|3(3) - 4(1) + 1|}{\sqrt{3^2 + (-4)^2}} = \frac{|9 - 4 + 1|}{5} = \frac{|6|}{5} = \frac{6}{5} \]

### Step 4: Identify the radii of the circles
Since the line is a tangent to both circles, the distances calculated above represent the radii of the circles:
- Radius of circle \( C_1 \) is \( R_1 = \frac{4}{5} \)
- Radius of circle \( C_2 \) is \( R_2 = \frac{6}{5} \)

### Step 5: Analyze the statements
Now we need to analyze the statements regarding the tangents and the radii:
1. **Statement A**: \( L_1 \) is a Direct Common Tangent of Circles.
2. **Statement B**: \( L_1 \) is a Transverse Common Tangent of Circles.
3. **Statement C**: Radius of \( C_1 \) is \( \frac{4}{5} \) units.
4. **Statement D**: Radius of \( C_2 \) is \( \frac{6}{5} \) units.

### Step 6: Determine the incorrect statement
From our analysis:
- Since the line \( L_1 \) touches the circles on opposite sides, it is a **Transverse Common Tangent** (Statement B is correct).
- The radii calculated are correct (Statements C and D are correct).
- Therefore, **Statement A** is incorrect.

### Final Answer
The incorrect statement is:
**A. \( L_1 \) is a Direct Common Tangent of Circles.**