The value of k so that `(x-1)/(-3) = (y-2)/(2k) = (z-4)/(2) and (x-1)/(3k) = (y-1)/(1) = (z-6)/(-5)` may be perpendicular is given by :

Options

A. `- 7/10`
B. `-10/7`
C. `-10`
D. `10/7`

Answer

To find the value of \( k \) such that the given lines are perpendicular, we will follow these steps:

### Step 1: Identify the Direction Vectors
The given equations are:
1. \(\frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-4}{2}\)
2. \(\frac{x-1}{3k} = \frac{y-1}{1} = \frac{z-6}{-5}\)

From these equations, we can extract the direction vectors.

For the first line, the direction vector is:
\[ \mathbf{d_1} = (-3, 2k, 2) \]

For the second line, the direction vector is:
\[ \mathbf{d_2} = (3k, 1, -5) \]

### Step 2: Use the Dot Product to Determine Perpendicularity
Two vectors are perpendicular if their dot product is zero. Therefore, we need to calculate:
\[ \mathbf{d_1} \cdot \mathbf{d_2} = 0 \]

Calculating the dot product:
\[ (-3)(3k) + (2k)(1) + (2)(-5) = 0 \]

### Step 3: Simplify the Dot Product Equation
Expanding the dot product gives:
\[ -9k + 2k - 10 = 0 \]

Combining like terms:
\[ -7k - 10 = 0 \]

### Step 4: Solve for \( k \)
Now, isolate \( k \):
\[ -7k = 10 \\ k = -\frac{10}{7} \]

### Final Answer
The value of \( k \) such that the lines are perpendicular is:
\[ k = -\frac{10}{7} \]

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