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 find the value of \( k \) such that the line

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

is one of the angle bisectors of the lines

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

and

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

we will follow these steps:

### Step 1: Identify the direction ratios of the given lines

The direction ratios of the first line \( l_1 \) can be derived from the equation \( \frac{x}{1} = \frac{y}{-2} = \frac{z}{3} \). Thus, the direction ratios are \( (1, -2, 3) \).

The direction ratios of the second line \( l_2 \) from the equation \( \frac{x}{-2} = \frac{y}{3} = \frac{z}{1} \) are \( (-2, 3, 1) \).

### Step 2: Find the angle bisectors

The angle bisectors of two lines with direction ratios \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \) can be found using the formula:

\[ \frac{(a_1, b_1, c_1)}{\sqrt{a_1^2 + b_1^2 + c_1^2}} \pm \frac{(a_2, b_2, c_2)}{\sqrt{a_2^2 + b_2^2 + c_2^2}} \]

Calculating the magnitudes:

For \( l_1 \):
\[ \sqrt{1^2 + (-2)^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

For \( l_2 \):
\[ \sqrt{(-2)^2 + 3^2 + 1^2} = \sqrt{4 + 9 + 1} = \sqrt{14} \]

Thus, the angle bisectors are given by:

\[ \frac{(1, -2, 3)}{\sqrt{14}} \pm \frac{(-2, 3, 1)}{\sqrt{14}} \]

### Step 3: Calculate the angle bisectors

Calculating the first angle bisector:

\[ \frac{(1 - 2, -2 + 3, 3 + 1)}{\sqrt{14}} = \frac{(-1, 1, 4)}{\sqrt{14}} \]

Calculating the second angle bisector:

\[ \frac{(1 + 2, -2 - 3, 3 - 1)}{\sqrt{14}} = \frac{(3, -5, 2)}{\sqrt{14}} \]

### Step 4: Check if the given line is an angle bisector

The line given is

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

This line passes through the point \( (2, -2, -k) \). To check if it passes through the origin \( (0, 0, 0) \):

Setting \( x = 0, y = 0, z = 0 \):

From the first part of the equation:

\[ \frac{0-2}{-1} = 2 \]

From the second part:

\[ \frac{0+2}{1} = 2 \]

From the third part:

\[ \frac{0+k}{4} = \frac{k}{4} \]

Setting these equal gives:

\[ 2 = \frac{k}{4} \implies k = 8 \]

### Conclusion

Thus, the value of \( k \) is

\[ \boxed{8} \]