Question

If the lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{\lambda }$ and $\frac{x-1}{\lambda }=\frac{y-4}{2}=\frac{z-5}{1}$ intersect, then the value(s) of $\lambda$ can be

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

Answer: (A), (B)

$0$

Given line : $L_1:\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{\lambda }$ passes through point $A(2,3,4)$ and parallel to vector $\hat{i}+\hat{j}+\lambda \hat{k}$ and
$L_2:\frac{x-1}{\lambda }=\frac{y-4}{2}=\frac{z-5}{1}$
passes through point $B(1,4,5)$ and parallel to vector $\lambda \hat{i}+2\hat{j}+\hat{k}$
as we can see the given lines are not parallel,
so for intersecting line : vector $\overrightarrow{AB},\hat{i}+\hat{j}+\lambda \hat{k},\lambda \hat{i}+2\hat{j}+\hat{k}$ are coplanar
$\left|\begin{array}{ccc}2-1& 3-4& 4-5\\ 1& 1& \lambda \\ \lambda & 2& 1\end{array}\right|=0$
$\Rightarrow 1(1-2\lambda )+(1-{\lambda }^2)-(2-\lambda )=0\Rightarrow -\lambda -{\lambda }^2=0\Rightarrow \lambda =0,-1$