Question
If the line $$\frac{{x - 1}}{2} = \frac{{y + 1}}{3} = \frac{{z - 1}}{4}$$ and $$\frac{{x - 3}}{1} = \frac{{y - k}}{2} = \frac{z}{1}$$ intersect, then $$k$$ is equal to :
A.
$$-1$$
B.
$$\frac{2}{9}$$
C.
$$\frac{9}{2}$$
D.
$$0$$
Answer :
$$\frac{9}{2}$$
Solution :
Given lines in vector form are $$\vec r = \left( {\hat i - \hat j + \hat k} \right) + \lambda \left( {2\hat i + 3\hat j + 4\hat k} \right)$$
and $$\vec r = \left( {3\hat i + k\hat j} \right) + \mu \left( {\hat i + 2\hat j + \hat k} \right)$$
These will intersect if shortest distance between them $$=0$$
\[\begin{array}{l}
{\rm{i}}{\rm{.e}}{\rm{.,}}\,\,\,\,\left( {{{\vec a}_2} - {{\vec a}_1}} \right).{{\vec b}_1} \times {{\vec b}_2} = 0\\
\Rightarrow \left| \begin{array}{l}
3 - 1\,\,\,\,\,k + 1\,\,\,\,\, - 1\\
\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4\\
\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1
\end{array} \right| = 0\\
\Rightarrow 2\left( { - 5} \right) - \left( {k + 1} \right)\left( { - 2} \right) - 1\left( 1 \right) = 0\\
\Rightarrow k = \frac{9}{2}
\end{array}\]