Question

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then the value of $2k$ is

Answer 1

Any point on $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}=\lambda$ is

$(2\lambda +1,3\lambda -1,4\lambda +1),\lambda \in R$

Any point on $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}=\mu$ is

$(\mu +3,2\mu +k,\mu )$; $\mu \in R$

So,
$\begin{array}{c}2\lambda +1=\mu +3\\ 3\lambda -1=2\mu +k\\ 4\lambda +1=\mu \end{array}\}$

So, $\lambda =-\frac{3}{2}$, $\mu =-5\Rightarrow k=\frac{9}{2}$