Question

The line $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ 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

The correct option is C k = 0 or -3

$\left|\begin{array}{ccc}{x}_2-x_1& y_2-y_1& z_2-z_1\\ l_1& m_1& n_1\\ l_2& m_2& n_2\end{array}\right|=0$
$\left|\begin{array}{ccc}1& -1& -1\\ 1& 1& -k\\ k& 2& 1\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}0& 0& -1\\ 2& 1+k& -k\\ k+2& 1& 1\end{array}\right|=0$
$k^2+3k^2=0\Rightarrow k(k+3)=0\Rightarrow k=0or-3$.