If 2 , 3 , 9 , 5 , 2 , 1 , 1 , λ , 8 and λ , 2 , 3 are coplanar, then the product of all possible…

Given,

Points $A\left(2,3,9\right), B\left(5,2,1\right), C\left(1,\lambda ,8\right) \& D\left(\lambda ,2,3\right)$ are coplanar,

Now we know that condition of coplanarity is  $\left[\begin{array}{ccc}\overrightarrow{AB}& \overrightarrow{AC}& \overrightarrow{AD}\end{array}\right]=0$

$\Rightarrow \left|\begin{array}{ccc}3& -1& -8\\ -1& \lambda -3& -1\\ \lambda -2& -1& -6\end{array}\right|=0$

$\Rightarrow 3\left(\left(-6\right)\left(\lambda -3\right)-1\right)+1\left(6+\lambda -2\right)-8\left(1-\left(\lambda -3\right)\left(\lambda -2\right)\right)=0$

$\Rightarrow 3\left(-6\lambda +17\right)+4+\lambda -8+8\left(\lambda -3\right)\left(\lambda -2\right)=0$

$\Rightarrow -18\lambda +51+4+\lambda -8+8{\lambda }^2-40\lambda +48=0$

$\Rightarrow 8{\lambda }^2-57\lambda +95=0$

So product of roots will be $\frac{95}{8}$