The lines `(x+1)/(1)=(y-1)/(2)=(z-2)/(1),(x-1)/(2)=(y)/(1)=(z+1)/(4)` are

Options

A. parallel lines
B. intersecting lines
C. skew lines
D. None of the above

Answer

To determine the relationship between the two lines given by the equations:

1. \(\frac{x+1}{1} = \frac{y-1}{2} = \frac{z-2}{1}\)
2. \(\frac{x-1}{2} = \frac{y}{1} = \frac{z+1}{4}\)

we will follow these steps:

### Step 1: Identify Direction Ratios and Points
From the first line, we can express it in parametric form:
- Let \( t \) be the parameter.
- \( x = t - 1 \)
- \( y = 2t + 1 \)
- \( z = t + 2 \)

Thus, the direction ratios for the first line (let's call it \( L_1 \)) are \( (1, 2, 1) \) and a point on the line is \( (-1, 1, 2) \).

For the second line, we can express it in parametric form:
- Let \( s \) be the parameter.
- \( x = 2s + 1 \)
- \( y = s \)
- \( z = 4s - 1 \)

Thus, the direction ratios for the second line (let's call it \( L_2 \)) are \( (2, 1, 4) \) and a point on the line is \( (1, 0, -1) \).

### Step 2: Check for Parallelism
To check if the lines are parallel, we need to see if the direction ratios are proportional:
- Direction ratios of \( L_1 \): \( (1, 2, 1) \)
- Direction ratios of \( L_2 \): \( (2, 1, 4) \)

We check the ratios:
\[ \frac{1}{2}, \frac{2}{1}, \frac{1}{4} \]
These ratios are not equal, indicating that the lines are not parallel.

### Step 3: Check for Intersection
To check if the lines intersect, we need to find if there exist parameters \( t \) and \( s \) such that:
\[ t - 1 = 2s + 1 \quad (1) \]
\[ 2t + 1 = s \quad (2) \]
\[ t + 2 = 4s - 1 \quad (3) \]

From equation (2), we can express \( s \) in terms of \( t \):
\[ s = 2t + 1 \]

Substituting \( s \) into equations (1) and (3):
1. From (1):
\[ t - 1 = 2(2t + 1) + 1 \implies t - 1 = 4t + 2 + 1 \implies t - 1 = 4t + 3 \implies -3t = 4 \implies t = -\frac{4}{3} \]

2. From (3):
\[ t + 2 = 4(2t + 1) - 1 \implies t + 2 = 8t + 4 - 1 \implies t + 2 = 8t + 3 \implies -7t = 1 \implies t = -\frac{1}{7} \]

Since \( t \) yields different values in both equations, the lines do not intersect.

### Conclusion
Since the lines are neither parallel nor intersecting, they are skew lines.

### Final Answer
The lines are skew lines.