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

Answer 1

Answer: (C)

$k=0$ or $-3$

For two lines to be coplanar,

$\left|\begin{array}{ccc}{x}_2-x_1& y_2-y_1& z_2-z_1\\ a_1& b_1& c_1\\ a_2& b_2& c_2\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}-1& 1& 1\\ 1& 1& -k\\ k& 2& 1\end{array}\right|=0$

$\Rightarrow -1(1+2k)-1(1+k^2)+1(2-k)=0$
$\Rightarrow k^2+3k=0$

$\Rightarrow k=0,-3$

Hence, option 'C' is correct.