If the lines \(\displaystyle\frac{1-x}{3}=\frac{y-2}{2\lambda}=\frac{z-3}{2}\) and \(\displaystyle\frac{x-1}{3\lambda}=\frac{y-1}{1}=\frac{6-z}{5}\) are perpendicular, then \(\lambda\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{-10}{7}\)
The direction ratios are read only after each line is written in the standard form \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\), with x, y, z having coefficient +1:
\( \displaystyle \frac{x-1}{-3} = \frac{y-2}{2\lambda} = \frac{z-3}{2}:\quad (-3,\ 2\lambda,\ 2) \)
\( \displaystyle \frac{x-1}{3\lambda} = \frac{y-1}{1} = \frac{z-6}{-5}:\quad (3\lambda,\ 1,\ -5) \)
For perpendicular lines, \(a_1a_2 + b_1b_2 + c_1c_2 = 0\):
\((-3)(3\lambda) + (2\lambda)(1) + (2)(-5) = 0\)
\(-9\lambda + 2\lambda - 10 = 0\)
\(-7\lambda = 10\)
\( \displaystyle \lambda = \frac{-10}{7} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{10}{11}\) — Incorrect. This comes from taking (3, 2λ, 2) for the first line, missing the minus sign of \(1 - x\): \(9\lambda + 2\lambda - 10 = 0\).
Option (3) \(\displaystyle\frac{-10}{11}\) — Incorrect. This comes from reading both sets of denominators as printed, (3, 2λ, 2) and (3λ, 1, 5): \(9\lambda + 2\lambda + 10 = 0\).
Option (4) \(\displaystyle\frac{10}{7}\) — Incorrect. This comes from taking (3λ, 1, 5) for the second line, missing the minus sign of \(6 - z\): \(-9\lambda + 2\lambda + 10 = 0\).