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  • If <img alt="\mathrm{(2,3,9),(5,2,1),(1, \lambda, 8) \: and\: (\lambda, 2,3)}" src="/latex-image/?%5Cmathrm%7B%282%2C3%2C9%29%2C%285%2C2%2C1%29%2C%281%2C%20%5Clambda%2C%208%29%20%5C%3A%20and%5C%3A%

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

Option: 1

\frac{21}{2}


Option: 2

\frac{59}{8}


Option: 3

\frac{57}{8}


Option: 4

\frac{95}{8}


Answers (1)

best_answer

\begin{aligned} & \text{A, B, C, D are coplanar implies } \mathrm{ \overrightarrow{A B}, \overrightarrow{A C}, \overrightarrow{A D}}\text{ are coplanar}\\ &\Rightarrow\left[\begin{array}{lll}\mathrm{ \overrightarrow{A B} }& \mathrm{ \overrightarrow{A C}} & \mathrm{ \overrightarrow{A D}}\end{array}\right]=0 \\ & \Rightarrow\left|\begin{array}{ccc}3 & -1 & -8 \\-1 & \lambda-3 & -1 \\2-2 & -1 & -6\end{array}\right|=0 \\ & \mathrm{ \Rightarrow 3[-6(\lambda-3)-1]+1[6+\lambda-2] }\\ &\mathrm{ -8[1-(2-2)(\lambda-3)]=0} \\ & \mathrm{ \Rightarrow 3[-6 h+17]+4+h-8\left[1-\left(h^{2}-5 h+6\right)\right]} \\ & \mathrm{ \Rightarrow-18 \lambda+51+4+\lambda-8\left[-\lambda^{2}+5 \lambda-5\right]=0} \\ & \mathrm{ \Rightarrow-17 \lambda+55+8 \lambda^{2}-40 \lambda+40=0 }\\ &\mathrm{ \Rightarrow 8 \lambda^{2}-57 \lambda+95=0}\\ & \text{Product }\mathrm{ R_{1} R_{2}=\frac{95}{8}}\\ &\therefore \text{option (D)} \end{aligned}

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vishal kumar

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