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

Answer 1

Answer: (B)

$k=0$ or $k=-3$

Using fact, two line $\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2}=\frac{y-y_2}{b_2}=\frac{z-z_2}{c_2}$ are coplanar if
$\left|\begin{array}{ccc}{x}_2-x_1& y_2-y_1& z_2-z_1\\ a_1& b_1& c_1\\ a_2& b_2& c_2\end{array}\right|=0$

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

$\Rightarrow k=0$ or $k=-3$