The lines \(\displaystyle\frac{x-1}{k}=\frac{y+2}{2}=\frac{3-z}{1}\) and \(\displaystyle\frac{x+3}{3}=\frac{y-1}{-k}=\frac{z-2}{4}\) are perpendicular to each other. The value of k is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 4
In the first line \(\frac{3-z}{1} = \frac{z-3}{-1}\), so its direction ratios are k, 2, −1. The second line has direction ratios 3, −k, 4.
The lines are perpendicular when \(a_1a_2+b_1b_2+c_1c_2 = 0\):
\((k)(3)+(2)(-k)+(-1)(4) = 0\)
\(3k-2k-4 = 0\)
\(k = 4\)
Why the other options are incorrect
Option (2) −4 — Incorrect. This comes from taking the third direction ratio of the first line as +1 instead of −1: \(3k-2k+4 = 0\).
Option (3) 2 — Incorrect. With k = 2 the sum is \(6-4-4 = -2 \ne 0\).
Option (4) 6 — Incorrect. With k = 6 the sum is \(18-12-4 = 2 \ne 0\).