JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank Plane

  • question_answer
    If the distance of the point (1, 1,1) from the origin is half its distance from the plane \[x+y+z+k=0\], then \[k=\]                              [Kerala (Engg.)2005]

    A)            \[\pm 3\]

    B)            \[\pm 6\]

    C)            ?3, 9

    D)            \[3,\,-9\]                                   

    Correct Answer: D

    Solution :

               Distance of the point (1,1,1) from origin                     \[=\sqrt{{{(1)}^{2}}+{{(1)}^{2}}+{{(1)}^{2}}}=\sqrt{3}\]                    Distance of the point (1,1,1) from \[x+y+z+k=0\] is, \[\pm \frac{(1)+(1)+(1)+k}{\sqrt{{{(1)}^{2}}+{{(1)}^{2}}+{{(1)}^{2}}}}=\pm \frac{k+3}{\sqrt{3}}\]            According to question, \[\sqrt{3}=\pm \frac{1}{2}\left( \frac{k+3}{\sqrt{3}} \right)\]            Þ \[6=\pm (k+3)\]Þ\[k=3,-9\].                 3, 9                                       

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