JEE Main 2023 (Online) 6th April Morning ShiftQuestion 1
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 $S=\{\lambda\in\mathbb{R}$: the points $A,B,C$ and $D$ are coplanar$\}$. Then $\sum_{\lambda\in S}(\lambda+2)^{2}$ is equal to
Vector Algebra
Solution
Coplanar: $\det[\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AD}]=0$ with $\overrightarrow{AB}=(-4,-3,3-2\lambda)$, $\overrightarrow{AC}=(-7,\lambda-5,4-2\lambda)$, $\overrightarrow{AD}=(-6,0,6-2\lambda)$. Expanding gives $\lambda^2-5\lambda+6=0\Rightarrow\lambda=2,3$. $\sum(\lambda+2)^2=16+25=41$. Option (C).