JEE Main & Advanced AIEEE Solved Paper-2006

  • question_answer
    The value 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     AIEEE  Solved  Paper-2006

    A) -2 and -1         

    B)        -2 and 1               

    C)                        2 and -1          

    D)        2 and 1

    Correct Answer: D

    Solution :

    Since, position vectors of A,B, C are\[2\hat{i}-\hat{j}+\hat{k},\] \[\hat{i}-3\hat{j}-5\hat{k}\] and\[a\hat{i}-3\hat{j}+\hat{k},\]respectively. \[\therefore \]\[AC=(a\hat{i}-3\hat{j}+\hat{k})-(2\hat{i}-\hat{j}+\hat{k})=(a-2)\hat{i}-2\hat{j}\] and \[BC=(a\hat{i}-3\hat{j}+\hat{k})-(\hat{i}-3\hat{j}-5\hat{k})\] \[=(a-1)\hat{i}+6\hat{k}\] Since, the \[\Delta ABC\] is right angled at C, then \[AC.BC=0\] \[\Rightarrow \]\[\{(a-2)\hat{i}-2\hat{j}\}.\{(a-1)\hat{i}+6\hat{k}\}=0\] \[\Rightarrow \]\[(a-2)(a-1)=0\]\[\Rightarrow \]\[a=1\] and 2

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