Let the position vectors of the points A , B , C and D be 5 i+5 j… - Vidyadip

MCQ

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

✓
$41$
B
$25$
C
$13$
D
$\frac{37}{2}$

✓

Answer

Correct option: A.

$41$

a
Since $A, B, C, D$ are coplanner

Hence $\left[\begin{array}{lll}\overrightarrow{ BA } & \overrightarrow{ CA } & \overrightarrow{ DA }\end{array}\right]=0$

$\begin{aligned}& \left|\begin{array}{ccc}4 & 3 & 2 \lambda-3 \\7 & 5-\lambda & 2 \lambda-4 \\6 & 0 & 2 \lambda-6\end{array}\right|=0 \\& \lambda=2,3 \text { Hence } \sum_{\lambda \in S}(\lambda+2)^2=41\end{aligned}$

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