The lines `x/1=y/2=z/3` and `(x-1)/(-2)=(y-2)/(-4)=(z-3)/(-6)` are

Options

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

Explanation

To determine the relationship between the lines given by the equations \( \frac{x}{1} = \frac{y}{2} = \frac{z}{3} \) and \( \frac{x-1}{-2} = \frac{y-2}{-4} = \frac{z-3}{-6} \), we will follow these steps:

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

For the first line, \( \frac{x}{1} = \frac{y}{2} = \frac{z}{3} \), the direction ratios can be directly read as:
- \( b_1 = (1, 2, 3) \)

For the second line, \( \frac{x-1}{-2} = \frac{y-2}{-4} = \frac{z-3}{-6} \), the direction ratios are:
- \( b_2 = (-2, -4, -6) \)

### Step 2: Check if the direction ratios are proportional

To check if the lines are parallel, we need to see if the direction ratios \( b_1 \) and \( b_2 \) are proportional. This means we need to find a scalar \( k \) such that:
\[ b_1 = k \cdot b_2 \]

Calculating the proportionality:
- If we take \( k = -\frac{1}{2} \), then:
\[ k \cdot b_2 = -\frac{1}{2} \cdot (-2, -4, -6) = (1, 2, 3) = b_1 \]

Since \( b_1 \) is proportional to \( b_2 \), we conclude that the lines are parallel.

### Step 3: Conclusion

Since the direction ratios of the two lines are proportional, we can conclude that the lines are parallel.

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

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