Question

If $(1,5,35),(7,5,5),(1,\lambda ,7)$ and $(2\lambda ,1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is :

• A. $-\frac{44}{5}$
• B. $\frac{39}{5}$
• C. $-\frac{39}{5}$
• D. $\frac{44}{5}$

Answer 1

The correct option is D $\frac{44}{5}$

Let $P(1,5,35),Q(7,5,5),R(1,\lambda ,7),S(2\lambda ,1,2)$
Condition of coplanarity is,
$\left[\overrightarrow{PQ}\overrightarrow{PR}\overrightarrow{PS}\right]=0$
$\Rightarrow \left|\begin{array}{ccc}6& 0& -30\\ 0& \lambda -5& -28\\ 2\lambda -1& -4& -33\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{ccc}1& 0& -5\\ 0& \lambda -5& -28\\ 2\lambda -1& -4& -33\end{array}\right|=0$

$\Rightarrow (-33\lambda +165-112)+5(\lambda -5)(2\lambda -1)=0$
$\Rightarrow 53-33\lambda +5(2{\lambda }^2-11\lambda +5)=0$
$\Rightarrow 10{\lambda }^2-88\lambda +78=0$
$\Rightarrow 5{\lambda }^2-44\lambda +39=0$
$\therefore {\lambda }_1+{\lambda }_2=\frac{44}{5}$