A) \[k=0\]or ?1
B) \[k=0\]or 1
C) \[k=0\]or ?3
D) \[k=3\]or ?3
Correct Answer: C
Solution :
\[\left| \,\begin{matrix} {{x}_{2}}-{{x}_{1}} & {{y}_{2}}-{{y}_{1}} & {{z}_{2}}-{{z}_{1}} \\ {{l}_{1}} & {{m}_{1}} & {{n}_{1}} \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}} \\ \end{matrix}\, \right|\,\,=\,\,0\] \[\left| \text{ }\begin{matrix} 1 & -1 & -1 \\ 1 & 1 & -k \\ k & 2 & 1 \\ \end{matrix}\text{ } \right|=0\Rightarrow \left| \text{ }\begin{matrix} 0 & 0 & -1 \\ 2 & 1+k & -k \\ k+2 & 1 & 1 \\ \end{matrix}\text{ } \right|\,=\,0\] \[{{k}^{2}}+3{{k}^{2}}=0\Rightarrow k(k+3)=0\]Þ \[k=0\,\,\,\text{or }-3\].
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