If the lines \( \displaystyle\frac{x-1}{-3}=\frac{y-2}{2\lambda}=\frac{3-z}{-2} \) and \( \displaystyle\frac{x-1}{3\lambda}=\frac{1-y}{-1}=\frac{6-z}{5} \) are perpendicular to each other, then the value of \( \lambda \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → -10/7
Direction ratios can be read only when x, y and z each appear with coefficient +1 in the numerators, so both lines are first written in that form.
First line: \( \frac{3-z}{-2}=\frac{z-3}{2} \), so
\( \displaystyle \frac{x-1}{-3}=\frac{y-2}{2\lambda}=\frac{z-3}{2} \)
\( \displaystyle \text{direction ratios}\ -3,\ 2\lambda,\ 2 \)
Second line: \( \frac{1-y}{-1}=\frac{y-1}{1} \) and \( \frac{6-z}{5}=\frac{z-6}{-5} \), so
\( \displaystyle \frac{x-1}{3\lambda}=\frac{y-1}{1}=\frac{z-6}{-5} \)
\( \displaystyle \text{direction ratios}\ 3\lambda,\ 1,\ -5 \)
For perpendicular lines \( a_1a_2+b_1b_2+c_1c_2=0 \):
\( \displaystyle (-3)(3\lambda)+(2\lambda)(1)+(2)(-5)=0 \)
\( -9\lambda+2\lambda-10=0 \)
\( -7\lambda=10 \)
\( \displaystyle \lambda=-\frac{10}{7} \)
Why the other options are incorrect
Option (2) 7/10 — Incorrect. With \( \lambda=\frac{7}{10} \), \( -7\lambda-10=-17\ne 0 \), so the lines are not perpendicular.
Option (3) 10/7 — Incorrect. This comes from reading the last direction ratio of the first line as −2 straight from \( \frac{3-z}{-2} \), without first making the coefficient of z equal to +1: then \( -9\lambda+2\lambda+10=0 \) gives \( \frac{10}{7} \).
Option (4) -7/10 — Incorrect. With \( \lambda=-\frac{7}{10} \), \( -7\lambda-10=-3\ne 0 \), so the lines are not perpendicular.