If the lines `(x-1)/(-3)=(y-2)/(2k)=(z-3)/2`and `(x-1)/(3k)=(y-1)/1=(z-6)/(-5)`are perpendicular, find the value of `k.`

Options

A. `-11/19`
B. `-8/7`
C. `-10/7`
D. `-9/7`

Explanation

To solve the problem, we need to find the value of \( k \) such that the two given lines are perpendicular. The lines are represented in symmetric form, and we can extract their direction ratios from these equations.

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

The first line is given by:
\[ \frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2} \]
From this, we can identify the direction ratios as:
\[ \mathbf{d_1} = (-3, 2k, 2) \]

The second line is given by:
\[ \frac{x-1}{3k} = \frac{y-1}{1} = \frac{z-6}{-5} \]
From this, we can identify the direction ratios as:
\[ \mathbf{d_2} = (3k, 1, -5) \]

### Step 2: Use the condition for perpendicularity

For two lines to be perpendicular, the dot product of their direction ratios must be zero. Thus, we have:
\[ \mathbf{d_1} \cdot \mathbf{d_2} = 0 \]
Calculating the dot product:
\[ (-3)(3k) + (2k)(1) + (2)(-5) = 0 \]
This simplifies to:
\[ -9k + 2k - 10 = 0 \]

### Step 3: Solve for \( k \)

Combining like terms:
\[ -7k - 10 = 0 \]
Adding 10 to both sides:
\[ -7k = 10 \]
Dividing by -7:
\[ k = -\frac{10}{7} \]

### Final Answer
Thus, the value of \( k \) for which the given lines are perpendicular is:
\[ \boxed{-\frac{10}{7}} \]