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. $3$
• B. $4$
• C. $1$
• D. $2$

Answer 1

The correct option is A $3$

Given lines are coplanar.
$\Rightarrow \left|\begin{array}{ccc}3-1& 2-2& 1-(-3)\\ 1& 2& {\lambda }^2\\ 1& {\lambda }^2& 2\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{ccc}2& 0& 4\\ 1& 2& {\lambda }^2\\ 1& {\lambda }^2& 2\end{array}\right|=0$
$\Rightarrow 2(4-{\lambda }^4)+4({\lambda }^2-2)=0$
$\Rightarrow 2{\lambda }^4-4{\lambda }^2=0$
$\Rightarrow \lambda =0,\pm \sqrt{2}$