Total number of lines touching atleast two circles of the family of four circles `x^(2) + y^(2) +- 8x +- 8y = 0` is

Options

A. 8
B. 10
C. 12
D. 14

Answer

To find the total number of lines touching at least two circles of the family of four circles given by the equations \(x^2 + y^2 \pm 8x \pm 8y = 0\), we will first rewrite the equations of the circles, determine their centers and radii, and then count the number of common tangents between each pair of circles.

### Step 1: Rewrite the Circle Equations
The given equations can be rewritten by completing the square.

1. For the first circle:
\[ x^2 + y^2 - 8x + 8y = 0 \]
Completing the square:
\[ (x - 4)^2 + (y + 4)^2 = 32 \]
Center: \(C_1(4, -4)\), Radius: \(4\sqrt{2}\)

2. For the second circle:
\[ x^2 + y^2 - 8x - 8y = 0 \]
Completing the square:
\[ (x - 4)^2 + (y + 4)^2 = 32 \]
Center: \(C_2(4, 4)\), Radius: \(4\sqrt{2}\)

3. For the third circle:
\[ x^2 + y^2 + 8x - 8y = 0 \]
Completing the square:
\[ (x + 4)^2 + (y - 4)^2 = 32 \]
Center: \(C_3(-4, 4)\), Radius: \(4\sqrt{2}\)

4. For the fourth circle:
\[ x^2 + y^2 + 8x + 8y = 0 \]
Completing the square:
\[ (x + 4)^2 + (y + 4)^2 = 32 \]
Center: \(C_4(-4, -4)\), Radius: \(4\sqrt{2}\)

### Step 2: Analyze the Circles
The centers of the circles are:
- \(C_1(4, -4)\)
- \(C_2(4, 4)\)
- \(C_3(-4, 4)\)
- \(C_4(-4, -4)\)

All circles have the same radius \(r = 4\sqrt{2}\).

### Step 3: Count the Common Tangents
For any two circles, the number of common tangents can be determined based on their relative positions.

1. **Circles \(C_1\) and \(C_2\)**:
- Distance between centers = \(8\)
- Number of common tangents = 2 (external tangents)

2. **Circles \(C_1\) and \(C_3\)**:
- Distance between centers = \(8\)
- Number of common tangents = 2 (external tangents)

3. **Circles \(C_1\) and \(C_4\)**:
- Distance between centers = \(8\)
- Number of common tangents = 2 (external tangents)

4. **Circles \(C_2\) and \(C_3\)**:
- Distance between centers = \(8\)
- Number of common tangents = 2 (external tangents)

5. **Circles \(C_2\) and \(C_4\)**:
- Distance between centers = \(8\)
- Number of common tangents = 2 (external tangents)

6. **Circles \(C_3\) and \(C_4\)**:
- Distance between centers = \(8\)
- Number of common tangents = 2 (external tangents)

### Step 4: Total Count of Lines
Since each pair of circles contributes 2 common tangents, and there are 6 pairs of circles, the total number of lines touching at least two circles is:
\[ \text{Total Lines} = 6 \times 2 = 12 \]

### Final Answer
The total number of lines touching at least two circles of the given family is **12**.