Q.The lines $\dfrac{x-2}{1}=\dfrac{y-3}{1}=\dfrac{z-4}{-k}$ and $\dfrac{x-1}{k}=\dfrac{y-4}{2}=\dfrac{z-5}{1}$ are coplanar if
(A) k = 1 or -1
(B) k = 0 or -3
(C) k = ±3
(D) k = 0 or -1

Two parallel lines are always coplanar. For two non-parallel lines, coplanarity is equivalent to the shortest distance between them being zero — if that distance is not zero, the lines are skew instead. This gives the test: lines $\vec r=\vec a_1+\lambda_1\vec b_1$ and $\vec r=\vec a_2+\lambda_2\vec b_2$ are coplanar if and only if $(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)=0$. In Cartesian symmetric form, the same condition becomes a $3\times3$ determinant, built from the components of the connecting vector between the two given points over the two sets of direction ratios, being equal to zero. Once two lines are shown to be coplanar, the common plane they determine ca …