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10th Class Mathematics Coordinate Geometry Question Bank Coordinate Geometry

  • question_answer
    If the centroid of the triangle formed by the points (a, b), (b, c) and (c. a) is at the origin, then \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}=\]

    A)  \[abc\]                        

    B)  0                    

    C) \[a+b+c\]     

    D)         \[3\text{ }abc\]

    Correct Answer: D

    Solution :

                                                Since centroid of triangle formed by  \[(a,\,b),\,(b,c)\] and \[(c,\,a)\] is at the origin. \[\therefore \]    \[\frac{a+b+c}{3}=0\,\,\,\,\Rightarrow \,\,a+b+c=0\]     ?.(i) Also,   \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=\frac{1}{2}[a+b+c]\]             \[[{{(a-b)}^{2}}{{(b-c)}^{2}}+{{(c-a)}^{2}}]\] \[\Rightarrow \]            \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=0\]      (from (i)) \[\Rightarrow \]            \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc\]


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