Let the position vectors of the points A, B, C and D be 5 i ^ + 5 j ^ + 2 k ^ , i ^ + 2 j ^ + 3 k ^ , - 2 i ^ + j ^ + 4 k ^ and - i ^ + 5 j ^ + 6 k ^ . Let the set S = { : the points A, B, C and D are coplanar}. Then S ( + 2 ) 2 is equal t

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Solution

Q.
Let the position vectors of the points A, B, C and D be $5 \hat{i} + 5 \hat{j} + 2 λ \hat{k}, \hat{i} + 2 \hat{j} + 3 \hat{k}, - 2 \hat{i} + λ \hat{j} + 4 \hat{k}$ and $- \hat{i} + 5 \hat{j} + 6 \hat{k}$ . Let the set $S =$ { $λ \in ℝ :$ the points A, B, C and D are coplanar}. Then $\sum_{λ \in S} {(λ + 2)}^{2}$ is equal to [2023]

1 $\frac{37}{2}$

2 $25$

3 $13$

4 $41$

Ans.
(4)

We have, $\vec{A B} = - 4 \hat{i} - 3 \hat{j} + (3 - 2 λ) \hat{k}$

$\vec{A C} = - 7 \hat{i} + (λ - 5) \hat{j} + (4 - 2 λ) \hat{k}$

$\vec{A D} = - 6 \hat{i} + (6 - 2 λ) \hat{k}$

Since A, B, C and D are coplanar

$∴$ $| \begin{matrix} - 4 & - 3 & 3 - 2 λ \\ - 7 & λ - 5 & 4 - 2 λ \\ - 6 & 0 & 6 - 2 λ \end{matrix} |$ $= 0$

$\Rightarrow - 4 [(λ - 5) (6 - 2 λ) - 0] + 3 [- 7 (6 - 2 λ) + 6 (4 - 2 λ)]$ $+ (3 - 2 λ) [0 + 6 (λ - 5)] = 0$

$\Rightarrow - 4 (16 λ - 2 λ^{2} - 30) + 3 (2 λ - 18) + (3 - 2 λ) (6 λ - 30) = 0$

$\Rightarrow λ^{2} - 5 λ + 6 = 0 \Rightarrow λ = 2 or λ = 3$ $∴$ $S = {2, 3}$

So, $\sum_{λ \in S} {(λ + 2)}^{2} = 4^{2} + 5^{2} = 41$

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