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:
Solution
$\because A(2,3,9), B(5,2,1), C(1, \lambda, 8)$ and D$(\lambda, 2,3)$ are coplanar.
<br/><br/>$\therefore$ $[\overrightarrow{\mathrm{AB}} \,\,\,\,\overrightarrow{\mathrm{AC}} \,\,\,\, \overrightarrow{\mathrm{AD}}]=0$
<br/><br/>$\left|\begin{array}{ccc}3 & -1 & -8 \\ -1 & \lambda-3 & -1 \\ \lambda-2 & -1 & -6\end{array}\right|=0$
<br/><br/>$\Rightarrow[-6(\lambda-3)-1]-8(1-(\lambda-3)(\lambda-2))+(6+(\lambda$
$-2)=0$
<br/><br/>$\Rightarrow$ $3(-6 \lambda+17)-8\left(-\lambda^2+5 \lambda-5\right)+(\lambda+4)=8$
<br/><br/>$\Rightarrow$ $8 \lambda^2-57 \lambda+95=0$
<br/><br/>$\therefore$ $\lambda_1 \lambda_2=\frac{95}{8}$
About this question
Subject: Mathematics · Chapter: Three Dimensional Geometry · Topic: Direction Cosines and Ratios
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