Question

If for some $\alpha \in R ,$ the lines $L _{1}: \frac{ x +1}{2}=\frac{ y -2}{-1}=\frac{ z -1}{1}$ and $L _{2}: \frac{ x +2}{\alpha}=\frac{ y +1}{5-\alpha}=\frac{ z +1}{1}$ are coplanar, then the line $L_{2}$ passes through the point:

• A. $(-2,10,2)$
• B. $(10,2,2)$
• C. $(10,-2,-2)$
• D. $(2,-10,-2)$

Answer 1

Answer: (D)

$L _{1} \equiv \frac{ x +1}{2}=\frac{ y -2}{-1}=\frac{ z -1}{1}$
$L _{2} \equiv \frac{ x +2}{\alpha}=\frac{ y +1}{5-\alpha}=\frac{ z +1}{1}$
Point $A (-1,2,1) B (-2,-1,-1)$
$\because L _{1}$ and $L _{2}$ are coplanar
$\Rightarrow \begin{vmatrix}2&-1&1\\ \alpha&5-\alpha&1\\ 1&3&2\end{vmatrix} = 0$
$\alpha=-4$
$L _{2} \equiv \frac{ x +2}{-4}=\frac{ y +1}{9}=\frac{ z +1}{1}$
Check options (2,-10,-2) lies on $L_{2}$