Question

If the lines \[ \frac{1-x}{2} = \frac{y-8}{\lambda} = \frac{z-5}{2} \quad \text{and} \quad \frac{x-11}{5} = \frac{y-3}{3} = \frac{z-1}{1} \] are perpendicular, then \(\lambda =\)

• A. \(4\)
• B. \(-4\)
• C. \(\dfrac{8}{3}\)
• D. \(-\dfrac{8}{3}\)

Hint:

Two lines are perpendicular if the dot product of their direction ratios is zero.

Answer 1

The Correct Option is C

Step 1: Write direction ratios.
For the first line: \( \langle -2, \lambda, 2 \rangle \)
For the second line: \( \langle 5, 3, 1 \rangle \)

Step 2: Use perpendicularity condition.
If lines are perpendicular, dot product of direction ratios is zero: \[ (-2)(5) + (\lambda)(3) + (2)(1) = 0 \]
Step 3: Solve for \(\lambda\).
\[ -10 + 3\lambda + 2 = 0 \Rightarrow 3\lambda = 8 \Rightarrow \lambda = \frac{8}{3} \]