If the line `(x-2)/-1=(y+2)/1=(z+k)/4` is one of the angle bisector of the lines `x/1=y/-2=z/3` and `x/-2=y/3=z/1` then the value of k is

Options

A. 1
B. 2
C. 4
D. 8

Explanation

To solve the problem, we need to find the value of \( k \) such that the line given by

\[ \frac{x-2}{-1} = \frac{y+2}{1} = \frac{z+k}{4} \]

is one of the angle bisectors of the lines defined by

\[ \frac{x}{1} = \frac{y}{-2} = \frac{z}{3} \]

and

\[ \frac{x}{-2} = \frac{y}{3} = \frac{z}{1}. \]

### Step 1: Identify the point of intersection
Since the angle bisector must pass through the origin (0, 0, 0), we will substitute \( x = 0 \), \( y = 0 \), and \( z = 0 \) into the equation of the line.

### Step 2: Substitute the values into the line equation
Substituting \( x = 0 \), \( y = 0 \), and \( z = 0 \) into the line equation gives:

\[ \frac{0 - 2}{-1} = \frac{0 + 2}{1} = \frac{0 + k}{4}. \]

### Step 3: Simplify each part of the equation
Calculating each part:

1. The left side:
\[ \frac{-2}{-1} = 2. \]

2. The middle part:
\[ \frac{2}{1} = 2. \]

3. The right side:
\[ \frac{k}{4}. \]

Thus, we have:

\[ 2 = 2 = \frac{k}{4}. \]

### Step 4: Set up the equation for \( k \)
From the equation \( \frac{k}{4} = 2 \), we can solve for \( k \):

\[ k = 2 \times 4 = 8. \]

### Step 5: Conclusion
The value of \( k \) is \( 8 \).

### Final Answer
Thus, the value of \( k \) is \( \boxed{8} \).

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