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JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank Critical Thinking

  • question_answer
    If \[P\equiv (0,\,1,\,0),Q\equiv (0,\,0,\,1)\], then  projection of \[PQ\] on the plane \[x+y+z=3\]is [EAMCET 2002]

    A) \[\sqrt{3}\]

    B) 3

    C) \[\sqrt{2}\]

    D) 2

    Correct Answer: C

    Solution :

    • Given plane is \[x+y+z-3=0\]. From point P and Q draw PM and QN perpendicular on the given plane and QR ^ MP. \[|MP|\,=\frac{0+1+0-3}{\sqrt{{{1}^{2}}+{{1}^{2}}+{{1}^{2}}}}=\frac{-2}{\sqrt{3}}\],   \[|NQ|\,=\frac{-2}{\sqrt{3}}\]                                      
    • \[|PQ|\,=\sqrt{{{(0-0)}^{2}}+{{(0-1)}^{2}}+{{(1-0)}^{2}}}=\sqrt{2}\]                   
    • \[|RP|=|MP|-|MR|\,=\,|MP|-|NQ|=0\]                   
    • \ \[|NM|\,=\,|QR|\,=\sqrt{P{{Q}^{2}}-R{{P}^{2}}}=\sqrt{{{(\sqrt{2})}^{2}}-0}\]\[=\,\sqrt{2}.\]


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