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 coplaner if:

• A. $k=1\text{or}4$
• B. $k=-2\text{or}-3$
• C. $k=0\text{or}3$
• D. $k=0\text{or}-3$

Answer 1

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

Given lines:
$\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$
Let $(a_1,b_1,c_1)=(1,1,-k)$ and $(a_2,b_2,c_2)=(k,2,1)$ be the d.r.'s of the given lines.
For two lines to be coplanar,
$\left|\begin{array}{ccc}{x}_2-x_1& y_2-y_1& z_2-z_1\\ a_1& b_1& c_1\\ a_2& b_2& c_2\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}-1& 1& 1\\ 1& 1& -k\\ k& 2& 1\end{array}\right|=0$
$\Rightarrow -1(1+2k)-1(1+k^2)+1(2-k)=0$
$\Rightarrow k^2+3k=0$
$\Rightarrow k=0\text{or}-3$