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 $k$ is

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

Answer 1

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

For the lines to be coplanar, we have $\left|\begin{array}{ccc}1& 1& -k\\ k& 2& 1\\ (2-1)& (3-4)& (4-5)\end{array}\right|=0$

$\therefore$ $\left|\begin{array}{ccc}1& 1& -k\\ k& 2& 1\\ 1& -1& -1\end{array}\right|=0$

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