JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank Plane

  • question_answer
    A variable plane is at a constant distance p from the origin and meets the axes in A, B and C. The locus of the centroid of the tetrahedron \[OABC\] is

    A)            \[{{x}^{-2}}+{{y}^{-2}}+{{z}^{-2}}=16{{p}^{-2}}\]                         

    B)            \[{{x}^{-2}}+{{y}^{-2}}+{{z}^{-2}}=16{{p}^{-1}}\]

    C)            \[{{x}^{-2}}+{{y}^{-2}}+{{z}^{-2}}=16\]

    D)            None of these

    Correct Answer: A

    Solution :

               Plane is\[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\], where \[p=\frac{1}{\sqrt{\sum\limits_{{}}^{{}}{\left( \frac{1}{{{a}^{2}}} \right)}}}\]                     or \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}=\frac{1}{{{p}^{2}}}\]                        ?..(i)                    Now according to equation, \[x=\frac{a}{4},\ \ y=\frac{b}{4},\ \ z=\frac{c}{4}\]                    Put the values of x, y, z in (i), we get the locus of the centroid of the tetrahedron.

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