Question

The lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar if-

• A. $k=1$ or $-1$
• B. $k=0$ or $-3$
• C. $k=3$ or $-3$
• D. $k=0$ or $-1$

Answer 1

The correct option is B $k=0$ or $-3$

For two lines to be coplaner,
$\left|\begin{array}{ccc}{x}_2-x_1& y_2-y_1& z_2-z_1\\ a_1& b_1& c_1\\ a_2& b_2& c_2\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}-1& 1& 1\\ 1& 1& -k\\ k& 2& 1\end{array}\right|=0\Rightarrow -1(1+2k)-1(1+k^2)+1(2-k)=0$
$\Rightarrow k^2+3k=0\Rightarrow k=0,-3$
Hence, option 'B' is correct.