The values of $a$, for which the points $A, B, C$ with position vectors $2 \hat{i}-\hat{j}+\hat{k},…

The values of $a$, for which the points $A, B, C$ with position vectors $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $a \hat{i}-3 \hat{j}+\hat{k}$ respectively are the vertices of a right-angled triangle with $C=\frac{\pi}{2}$ are
  1. 2 and 1
  2. −2 and −1
  3. −2 and 1
  4. 2 and −1

Solution

$\Rightarrow \overrightarrow{B A}=\hat{i}-2 \hat{j}+6 \hat{k}$ $\overrightarrow{C A}=(2-a) \hat{i}+2 \hat{j}$ $\overrightarrow{C B}=(1-a) \hat{i}-6 \hat{k}$ $\overrightarrow{C A} \cdot \overrightarrow{C B}=0 \Rightarrow(2-a)(1-a)=0$ $\Rightarrow a=2,1$

Asked in: JEE Main 2006

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