Consider the circles `C_(1)-=x^(2)+y^(2)-2x-4y-4=0andC_(2)-=x^(2)+y^(2)+2x+4y+4=0` and the line `L-=x+2y+2=0` then

Options

A. L is the radical axis of `C_(1)andC_(2)`
B. L is the common tangent of `C_(1)andC_(2)`
C. L is the common chord of `C_(1)andC_(2)`
D. L is perpendicular to the line joining centres of `C_(1)andC_(2)`

Answer

To solve the problem, we need to analyze the given circles and the line to determine if the line is the radical axis, common tangent, common chord, or perpendicular to the line joining the centers of the circles.

### Step 1: Write the equations of the circles and the line
The equations of the circles are:
- Circle \( C_1: x^2 + y^2 - 2x - 4y - 4 = 0 \)
- Circle \( C_2: x^2 + y^2 + 2x + 4y + 4 = 0 \)

The equation of the line is:
- Line \( L: x + 2y + 2 = 0 \)

### Step 2: Subtract the equations of the circles
To find the radical axis, we subtract the equation of \( C_2 \) from \( C_1 \):
\[ C_1 - C_2: (x^2 + y^2 - 2x - 4y - 4) - (x^2 + y^2 + 2x + 4y + 4) = 0 \]
This simplifies to:
\[ -4x - 8y - 8 = 0 \]
Dividing by -4 gives:
\[ x + 2y + 2 = 0 \]
This matches the equation of line \( L \), confirming that \( L \) is the radical axis of circles \( C_1 \) and \( C_2 \).

### Step 3: Find the centers and radii of the circles
For circle \( C_1 \):
- Coefficients: \( g = -1, f = -2, c = -4 \)
- Center \( C_1 = (-g, -f) = (1, 2) \)
- Radius \( r_1 = \sqrt{g^2 + f^2 - c} = \sqrt{(-1)^2 + (-2)^2 - (-4)} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \)

For circle \( C_2 \):
- Coefficients: \( g = 1, f = 2, c = 4 \)
- Center \( C_2 = (-g, -f) = (-1, -2) \)
- Radius \( r_2 = \sqrt{g^2 + f^2 - c} = \sqrt{(1)^2 + (2)^2 - (4)} = \sqrt{1 + 4 - 4} = \sqrt{1} = 1 \)

### Step 4: Calculate the distance between the centers
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{((-1) - 1)^2 + ((-2) - 2)^2} = \sqrt{(-2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]

### Step 5: Compare the distance with the sum of the radii
Sum of the radii:
\[ r_1 + r_2 = 3 + 1 = 4 \]
Since \( d = 2\sqrt{5} \) (approximately 4.47) is greater than \( r_1 + r_2 = 4 \), the circles are external to each other. Thus, \( L \) is indeed the radical axis.

### Step 6: Check if the line is perpendicular to the line joining the centers
The slope of line \( L \):
\[ L: x + 2y + 2 = 0 \implies 2y = -x - 2 \implies y = -\frac{1}{2}x - 1 \quad \text{(slope } m_1 = -\frac{1}{2}\text{)} \]

The slope of the line joining the centers \( C_1(1, 2) \) and \( C_2(-1, -2) \):
\[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 2}{-1 - 1} = \frac{-4}{-2} = 2 \]

### Step 7: Check if the slopes are perpendicular
Two lines are perpendicular if the product of their slopes is -1:
\[ m_1 \cdot m_2 = -\frac{1}{2} \cdot 2 = -1 \]
Thus, the line \( L \) is perpendicular to the line joining the centers of circles \( C_1 \) and \( C_2 \).

### Conclusion
- The line \( L \) is the radical axis of circles \( C_1 \) and \( C_2 \).
- The line \( L \) is also perpendicular to the line joining the centers of the circles.

### Final Answers:
1. **L is the radical axis of C1 and C2** (Correct)
2. **L is the common tangent of C1 and C2** (Incorrect)
3. **L is the common chord of C1 and C2** (Incorrect)
4. **L is perpendicular to the line joining C1 and C2** (Correct)