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Manipal Engineering Manipal Engineering Solved Paper-2009

  • question_answer
    If the circle\[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]intersects the hyperbola\[xy={{c}^{2}}\]in four points\[({{x}_{i}},\,\,{{y}_{i}})\], for \[i=1,\,\,2,\,\,3\]and 4, then\[{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}\] equals

    A)  0                                            

    B) \[c\]

    C) \[a\]                     

    D)        \[{{c}^{4}}\]

    Correct Answer: A

    Solution :

    Given,   \[{{x}^{2}}{{y}^{2}}={{c}^{4}}\] \[\Rightarrow \]               \[{{y}^{2}}({{a}^{2}}-{{y}^{2}})={{c}^{4}}\] \[\Rightarrow \]               \[{{y}^{4}}-{{a}^{2}}{{y}^{2}}+{{c}^{4}}=0\] Let\[{{y}_{1}},\,\,{{y}_{1}},\,\,{{y}_{3}}\]and\[{{y}_{4}}\]are the roots. \[\therefore \]  \[{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}=0\]


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