Question

If the straight lines $x =1+ s, y = -3- \lambda s, z =1+ \lambda s$ and $x = \frac{t}{2} , y = 1 + t , z = 2 - t ,$ with parameters s and t respectively, are co-planar, then $\lambda$ equals.

• A. 0
• B. -1
• C. $ - \frac{1}{2}$
• D. - 2

Answer 1

Answer: (D)

The given lines are
$x - 1 = \frac{y +3}{- \lambda} = \frac{z - 1}{\lambda} =s$ ....(1)
and $2x = y - 1 = \frac{z -2}{-1} = t$ ....(2)
The lines are coplanar, if
$\begin{vmatrix}0-1&1-\left(-3\right)&2-\left(1\right)\\ 1&-\lambda&\lambda\\ \frac{1}{2}&1&-1\end{vmatrix} = 0 $
$C_{2 } \to C_{2} + C_{3} ;$
$ \begin{vmatrix}1&-5&-1\\ 1&0&\lambda\\ \frac{1}{2}&0&-1\end{vmatrix} = 0 \Rightarrow 5\left(-1 - \frac{\lambda}{2}\right) = 0 \Rightarrow \lambda = - 2 $