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KVPY Sample Paper KVPY Stream-SX Model Paper-16

  • 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 \[ay\text{ }=\text{ }{{c}^{2}}\] the coordinate of the orthocenter of the APQ 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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