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

  • question_answer
    Two systems of rectangular axes have the same origin. If a plane cuts them at distance a, b, c and a', b', c' from the origin, then [AIEEE 2003]

    A) \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}+\frac{1}{b{{'}^{2}}}+\frac{1}{c{{'}^{2}}}=0\]

    B) \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}-\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}+\frac{1}{b{{'}^{2}}}-\frac{1}{c{{'}^{2}}}=0\]

    C) \[\frac{1}{{{a}^{2}}}-\frac{1}{{{b}^{2}}}-\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}-\frac{1}{b{{'}^{2}}}-\frac{1}{c{{'}^{2}}}=0\]

    D) \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}-\frac{1}{a{{'}^{2}}}-\frac{1}{b{{'}^{2}}}-\frac{1}{c{{'}^{2}}}=0\]

    Correct Answer: D

    Solution :

    • Equation of planes be \[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\] and \[\frac{x}{{{a}'}}+\frac{y}{{{b}'}}+\frac{z}{{{c}'}}=1\] (Perpendicular distance on plane from origin is same)           
    • \ \[\left| \frac{-1}{\sqrt{\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}}} \right|\,\,=\,\,\left| \frac{-1}{\sqrt{\frac{1}{{{{{a}'}}^{2}}}+\frac{1}{{{{{b}'}}^{2}}}+\frac{1}{{{{{c}'}}^{2}}}}} \right|\]           
    • \ \[\sum \frac{1}{{{a}^{2}}}-\sum \frac{1}{{{{{a}'}}^{2}}}=0\].


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