If the lines \( \displaystyle\frac{x-1}{2}=\displaystyle\frac{y+1}{\lambda-3}=\displaystyle\frac{z-2}{2} \) and \( x=0,\ \displaystyle\frac{2y-4}{-2}=\displaystyle\frac{z+1}{2-\lambda} \) are perpendicular, then value of \( \lambda \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{7}{3} \)
Direction ratios of the first line: \( (2,\ \lambda-3,\ 2) \).
In the second line the coefficient of y must be 1 before the direction ratio can be read:
\( \displaystyle \frac{2y-4}{-2}=\frac{y-2}{-1} \)
Since x = 0 on the whole line, its direction ratios are \( (0,\ -1,\ 2-\lambda) \).
The lines are perpendicular, so the sum of the products of the direction ratios is 0:
\( 2(0)+(\lambda-3)(-1)+2(2-\lambda)=0 \)
\( 7-3\lambda=0 \)
\( \displaystyle \lambda=\frac{7}{3} \)