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KVPY Sample Paper KVPY Stream-SX Model Paper-2

  • question_answer
    If \[P\,({{x}_{1}},\,\,{{y}_{1}}),\]\[Q\,({{x}_{2}},\,\,{{y}_{2}}),\]\[R\,({{x}_{3}},\,\,{{y}_{3}})\] and \[S\,({{x}_{4}},\,\,{{y}_{4}})\] are four cyclic points on a rectangular hyperbola \[xy={{c}^{2}}\] the coordinate of the orthocenter of the \[\Delta \,PQR\] are

    A) \[({{x}_{4}},\,\,-{{y}_{4}})\]                

    B) \[({{x}_{4}},\,\,{{y}_{4}})\]

    C) \[(-{{x}_{4}},\,\,-\,{{y}_{4}})\]             

    D) \[(-{{x}_{4}},\,\,{{y}_{4}})\]

    Correct Answer: C

    Solution :

    Centre of circle through P, Q, R and S can be given as \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}}{2},\,\,\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}}{2} \right)\]and centroid of triangle PQR is \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\,\,\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\] hence orthocenter is \[(-{{x}_{4}},\,\,-{{y}_{4}}).\]


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