If the lines \(\displaystyle\frac{x-5}{5\lambda+2}=\frac{2-y}{5}=\frac{1-z}{-1}\) and \(x=\displaystyle\frac{y+1/2}{2\lambda}=\frac{z-1}{3}\) are perpendicular, then the value of \(\lambda\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1
The direction ratios are read from the standard form, where x, y and z each have coefficient +1. Since \(\frac{2-y}{5} = \frac{y-2}{-5}\) and \(\frac{1-z}{-1} = \frac{z-1}{1}\), the first line is
\( \displaystyle \frac{x-5}{5\lambda+2} = \frac{y-2}{-5} = \frac{z-1}{1} \)
with direction ratios \(5\lambda + 2,\ -5,\ 1\).
The second line is
\( \displaystyle \frac{x}{1} = \frac{y + \frac{1}{2}}{2\lambda} = \frac{z-1}{3} \)
with direction ratios \(1,\ 2\lambda,\ 3\).
The lines are perpendicular when \(a_1a_2 + b_1b_2 + c_1c_2 = 0\):
\((5\lambda + 2)(1) + (-5)(2\lambda) + (1)(3) = 0\)
\(5 - 5\lambda = 0\)
\(\lambda = 1\)