Question

let $P(4,1,\lambda )$ and $Q(2,-1,\lambda )$ be two points. A line having direction ratios $1,-1,6$ is perpendicular to the plane passing through the origin, $P$ and $Q$, then $\lambda$ equals

• A. $-\frac{1}{2}$
• B. $\frac{1}{2}$
• C. $1$
• D. none of these

Answer 1

The correct option is A $-\frac{1}{2}$

By given point $P(4,1,\lambda )$ and $Q(2,-1,\lambda )$
$\left|\begin{array}{cc}1& \lambda \\ -1& \lambda \end{array}\right|,\left|\begin{array}{cc}\lambda & 4\\ \lambda & 2\end{array}\right|,\left|\begin{array}{cc}4& 1\\ 2& -1\end{array}\right|\Rightarrow (\lambda +\lambda ),(2\lambda -4\lambda ),(-4-2)\Rightarrow -2\lambda ,2\lambda ,6\Rightarrow -2\lambda =1,2\lambda =-1$
$\Rightarrow \lambda =-\frac{1}{2}$