Question

If $(2,3,9),(5,2,1),(1,\lambda ,8)$ and $(\lambda ,2,3)$ are coplanar, then the product of all possible values of $\lambda$ is :

• A. $\frac{21}{2}$
• B. $\frac{59}{8}$
• C. $\frac{57}{8}$
• D. $\frac{95}{8}$

Answer 1

Answer: (D)

$A(2,3,9);B(5,2,1);C(1,\lambda ,8);D(\lambda ,2,3)$
$[\overrightarrow{AB}\overrightarrow{AC}\overrightarrow{AD}]=0$
$\left|\begin{array}{ccc}3& -1& -8\\ -1& \lambda -3& -1\\ \lambda -2& -1& -6\end{array}\right|=0$
$\Rightarrow [-6(\lambda -3)-1]-8(1-(\lambda -3)(\lambda -2))+(6+(\lambda -2)=0$
$3(-6\lambda +17)-8\left(-{\lambda }^2+5\lambda -5\right)+(\lambda +4)=8$
$8{\lambda }^2-57\lambda +95=0$
${\lambda }_1{\lambda }_2=\frac{95}{8}$