Question

The number of distinct real values of $\lambda$ for which the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{{\lambda }^2}$ and $\frac{x-3}{1}=\frac{y-2}{{\lambda }^2}=\frac{z-1}{2}$ are coplanar is:

• A. $2$
• B. $4$
• C. $3$
• D. $1$

Answer 1

Answer: (C)

$3$

$\because$ lines are coplanar

$\therefore \left|\begin{array}{ccc}3-1& 2-2& 1-(-3)\\ 1& 2& {\lambda }^2\\ 1& {\lambda }^2& 2\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{ccc}2& 0& 4\\ 1& 2& {\lambda }^2\\ 1& {\lambda }^2& 2\end{array}\right|=0$

$\Rightarrow 2(4-{\lambda }^4)+4({\lambda }^2-2)=0$

$\Rightarrow 4-{\lambda }^4+2{\lambda }^2-4=0$

$\Rightarrow {\lambda }^2({\lambda }^2-2)=0$

$\Rightarrow \lambda =0,\sqrt{2},-\sqrt{2}$