Question

If the lines given by \( \frac{x-1}{2\lambda} = \frac{y-1}{5} = \frac{z-1}{2} \) and \( \frac{x+2}{\lambda} = \frac{y+3}{\lambda} = \frac{z+5}{1} \) are parallel, then the value of \( \lambda \) is

• A. \( \frac{-2}{5} \)
• B. \( \frac{2}{5} \)
• C. \( \frac{5}{2} \)
• D. \( \frac{-5}{2} \)

Hint:

For parallel lines, the direction ratios must be proportional. Always check the proportionality of the direction ratios to determine if the lines are parallel.

Answer 1

The Correct Option is D

Step 1: Understanding the condition for parallel lines.
For two lines to be parallel, their direction ratios must be proportional. We extract the direction ratios from both lines and set up the proportionality condition.

Step 2: Setting up the direction ratios.
For the first line, the direction ratios are \( (2\lambda, 5, 2) \), and for the second line, the direction ratios are \( (\lambda, \lambda, 1) \).

Step 3: Solving the proportionality equation.
By equating the corresponding direction ratios, we get the equation: \[ \frac{2\lambda}{\lambda} = \frac{5}{\lambda} = \frac{2}{1} \] Solving this equation gives \( \lambda = \frac{-5}{2} \).

Step 4: Conclusion.
Thus, the value of \( \lambda \) is \( \frac{-5}{2} \), which makes option (D) the correct answer.