If [(2,3,9),(5,2,1),(1, , 8)] and [(, 2,3)] are coplanar, then the product of all possible…

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:

Accepted Answer

$\because A(2,3,9), B(5,2,1), C(1, \lambda, 8)$ and D$(\lambda, 2,3)$ are coplanar.

$$ \therefore $$ $[\overrightarrow{\mathrm{AB}}  \,\,\,\,\overrightarrow{\mathrm{AC}} \,\,\,\, \overrightarrow{\mathrm{AD}}]=0$

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

$\Rightarrow[-6(\lambda-3)-1]-8(1-(\lambda-3)(\lambda-2))+(6+(\lambda$
$-2)=0$

$$ \Rightarrow $$ $3(-6 \lambda+17)-8\left(-\lambda^2+5 \lambda-5\right)+(\lambda+4)=8$

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

$$ \therefore $$ $\lambda_1 \lambda_2=\frac{95}{8}$

Solutions · 1