The lines \(\displaystyle\frac{x-1}{3}=\frac{y+2}{k}=\frac{z}{-6}\) and \(\displaystyle\frac{x+4}{-1}=\frac{y-3}{2}=\frac{z-1}{2}\) are parallel. The value of k is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → −6
The direction ratios are 3, k, −6 and −1, 2, 2. Lines are parallel when their direction ratios are proportional:
\( \displaystyle \frac{3}{-1} = \frac{k}{2} = \frac{-6}{2} \)
The first and last ratios are both −3, so
\( \displaystyle \frac{k}{2} = -3 \)
\(k = -6\)
Why the other options are incorrect
Option (1) 6 — Incorrect. \(\frac{6}{2} = 3\), while the other two ratios are −3.
Option (3) \(\displaystyle\frac{15}{2}\) — Incorrect. This is the value for perpendicular lines: \(-3+2k-12 = 0\).
Option (4) \(\displaystyle -\frac{3}{2}\) — Incorrect. \(\frac{-3/2}{2} = -\frac{3}{4}\), not −3.