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. $k=1\text{or}-1$
• B. $k=0\text{or}-3$
• C. $k=3\text{or}-3$
• D. $k=0\text{or}-1$

Answer 1

The correct option is B $k=0\text{or}-3$

$\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}...\left(1\right)$
$\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}...\left(2\right)$

The lines $\left(1\right)$ and $\left(2\right)$ are coplanar if
$\left|\begin{array}{ccc}{x}_2-x_1& y_2-y_1& z_2-z_1\\ l_1& m_1& n_1\\ l_2& m_1& n_2\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{ccc}1& -1& -1\\ 1& 1& -k\\ k& 2& 1\end{array}\right|=0$
$\Rightarrow k^2+3k=0$
$\Rightarrow k=0\text{or}-3$