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
$\left|\begin{array}{ccc}0-1& 1-\left(-3\right)& 2-\left(1\right)\\ 1& -\lambda & \lambda \\ \frac{1}{2}& 1& -1\end{array}\right|=0$
$C_2\to C_2+C_3;$
$\left|\begin{array}{ccc}1& -5& -1\\ 1& 0& \lambda \\ \frac{1}{2}& 0& -1\end{array}\right|=0\Rightarrow 5\left(-1-\frac{\lambda }{2}\right)=0\Rightarrow \lambda =-2$