Lines `(x-1)/1 = (y-1)/2 = (z-3)/0` and `(x-2)/0 = (y-3)/0 = (z-4)/1` are

Options

A. parallel
B. coincident
C. skew
D. perpendicular

Answer

To determine the relationship between the two lines given in the question, we will analyze their direction ratios and apply the necessary mathematical concepts step by step.

### Step-by-Step Solution:

1. **Identify the Direction Ratios of Line 1**:
The equation of Line 1 is given as:
\[ \frac{x-1}{1} = \frac{y-1}{2} = \frac{z-3}{0} \]
From this, we can extract the direction ratios of Line 1:
\[ \text{Direction Ratios of Line 1} = (1, 2, 0) \]

2. **Identify the Direction Ratios of Line 2**:
The equation of Line 2 is given as:
\[ \frac{x-2}{0} = \frac{y-3}{0} = \frac{z-4}{1} \]
From this, we can extract the direction ratios of Line 2:
\[ \text{Direction Ratios of Line 2} = (0, 0, 1) \]

3. **Check for Parallelism**:
Two lines are parallel if their direction ratios are proportional. We compare the direction ratios:
- For Line 1: \( (1, 2, 0) \)
- For Line 2: \( (0, 0, 1) \)

Since there is no constant \( k \) such that:
\[ (1, 2, 0) = k(0, 0, 1) \]
The lines are not parallel.

4. **Check for Perpendicularity**:
Two lines are perpendicular if the dot product of their direction ratios is zero. We calculate the dot product:
\[ \text{Direction Ratios of Line 1} \cdot \text{Direction Ratios of Line 2} = (1, 2, 0) \cdot (0, 0, 1) \]
This results in:
\[ 1 \cdot 0 + 2 \cdot 0 + 0 \cdot 1 = 0 \]
Since the dot product is zero, the lines are perpendicular.

5. **Conclusion**:
Since the lines are neither parallel nor coincident, and we have established that they are perpendicular, we conclude that the correct answer is that the lines are perpendicular.

### Final Answer:
The lines are **perpendicular**.