The lines `(x-1)/1=(y+1)/-1=z/2` and `x/2=(y-1)/-2=(z-1)/lambda` are parallel if `

Options

A. parallel if `lambda=4`
B. perpendicular if `lambda=-1`
C. coplanar if `lambda=4`
D. skew lines `lambda=5`

Explanation

To determine the value of \( \lambda \) for which the lines

\[ \frac{x-1}{1} = \frac{y+1}{-1} = \frac{z}{2} \]

and

\[ \frac{x}{2} = \frac{y-1}{-2} = \frac{z-1}{\lambda} \]

are parallel, we can follow these steps:

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

For the first line, the direction ratios can be extracted from the equation:

\[ \frac{x-1}{1} = \frac{y+1}{-1} = \frac{z}{2} \]

The direction ratios \( (a_1, b_1, c_1) \) are:

- \( a_1 = 1 \)
- \( b_1 = -1 \)
- \( c_1 = 2 \)

For the second line:

\[ \frac{x}{2} = \frac{y-1}{-2} = \frac{z-1}{\lambda} \]

The direction ratios \( (a_2, b_2, c_2) \) are:

- \( a_2 = 2 \)
- \( b_2 = -2 \)
- \( c_2 = \lambda \)

### Step 2: Set up the condition for parallel lines

For two lines to be parallel, the ratios of their direction ratios must be equal:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Substituting the values we found:

\[ \frac{1}{2} = \frac{-1}{-2} = \frac{2}{\lambda} \]

### Step 3: Solve the equations

From the first two ratios:

\[ \frac{1}{2} = \frac{-1}{-2} \quad \text{(This is true)} \]

Now, equate the first ratio with the third ratio:

\[ \frac{1}{2} = \frac{2}{\lambda} \]

Cross-multiplying gives:

\[ 1 \cdot \lambda = 2 \cdot 2 \]

This simplifies to:

\[ \lambda = 4 \]

### Conclusion

Thus, the value of \( \lambda \) for which the lines are parallel is

\[ \lambda = 4. \]