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JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank Self Evaluation Test - Introduction to Three Dimensional Geometry

  • question_answer
    If the sum of the squares of the distance of the point (x, y, z) from the points (a, 0, 0) and (-a, 0, 0) is \[2{{c}^{2}}\], then which one of the following is correct?

    A) \[{{x}^{2}}+{{a}^{2}}=2{{c}^{2}}-{{y}^{2}}-{{z}^{2}}\]

    B) \[{{x}^{2}}+{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\]

    C) \[{{x}^{2}}-{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\]

    D) \[{{x}^{2}}+{{a}^{2}}={{c}^{2}}+{{y}^{2}}+{{z}^{2}}\]

    Correct Answer: B

    Solution :

    [b] Let the point be P(x, y, z) and two points, (a, 0, 0) and (- a, 0, 0) be A and B As given in the problem, \[P{{A}^{2}}+P{{B}^{2}}=2{{c}^{2}}\]  so, \[{{(x+a)}^{2}}+{{(y-0)}^{2}}+{{(z-0)}^{2}}\]\[+{{(x-a)}^{2}}+{{(y-0)}^{2}}+{{(z-0)}^{2}}=2{{c}^{2}}\] or, \[{{(x+a)}^{2}}+{{y}^{2}}+{{z}^{2}}+{{(x-a)}^{2}}+{{y}^{2}}+{{z}^{2}}=2{{c}^{2}}\] \[\Rightarrow {{x}^{2}}+2a+{{a}^{2}}+{{y}^{2}}+{{z}^{2}}+{{x}^{2}}-2a+{{a}^{2}}\]\[+{{y}^{2}}+{{z}^{2}}=2{{c}^{2}}\] \[\Rightarrow 2({{x}^{2}}+{{y}^{2}}+{{z}^{2}}+{{a}^{2}})=2{{c}^{2}}\] \[\Rightarrow {{x}^{2}}+{{y}^{2}}+{{z}^{2}}+{{a}^{2}}={{c}^{2}}\] \[\Rightarrow {{x}^{2}}+{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\]


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