The value of \(\lambda\) so that the lines \(\displaystyle\frac{1-x}{3}=\frac{7y-14}{2\lambda}=\frac{z-3}{2}\) and \(\displaystyle\frac{7-7x}{3\lambda}=\frac{y-5}{1}=\frac{6-z}{5}\) are at right angle, is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{70}{11}\)
The direction ratios can be read only from the standard form \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\), where x, y and z each have coefficient 1.
First line:
\( \displaystyle \frac{x - 1}{-3} = \frac{y - 2}{\frac{2\lambda}{7}} = \frac{z - 3}{2} \)
Direction ratios: \(-3,\ \frac{2\lambda}{7},\ 2\).
Second line:
\( \displaystyle \frac{x - 1}{-\frac{3\lambda}{7}} = \frac{y - 5}{1} = \frac{z - 6}{-5} \)
Direction ratios: \(-\frac{3\lambda}{7},\ 1,\ -5\).
The lines are perpendicular when \(a_1a_2 + b_1b_2 + c_1c_2 = 0\):
\( \displaystyle (-3)\left(-\frac{3\lambda}{7}\right) + \frac{2\lambda}{7}(1) + 2(-5) = 0 \)
\( \displaystyle \frac{11\lambda}{7} = 10 \)
\( \displaystyle \lambda = \frac{70}{11} \)
Tip: Reading the denominators 3, \(2\lambda\), 2 and \(3\lambda\), 1, 5 straight from the printed form is a common trap: first make the coefficients of x, y and z equal to 1.
NCERT (2026-27 edition):
Angle Between Two Lines
