11th Class Mathematics Conic Sections Question Bank Critical Thinking

  • question_answer If a circle cuts a rectangular hyperbola \[xy={{c}^{2}}\] in A, B, C, D and the parameters of these four points be \[{{t}_{1}},\ {{t}_{2}},\ {{t}_{3}}\] and \[{{t}_{4}}\] respectively. Then [Kurukshetra CEE 1998]

    A)            \[{{t}_{1}}{{t}_{2}}={{t}_{3}}{{t}_{4}}\]                                        

    B)            \[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=1\]

    C)            \[{{t}_{1}}={{t}_{2}}\]               

    D)            \[{{t}_{3}}={{t}_{4}}\]

    Correct Answer: B

    Solution :

               Let equation of circle is \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]            Parametric form of \[xy={{c}^{2}}\] are \[x=ct,\,\,\,y=\frac{c}{t}\]            Þ \[{{c}^{2}}{{t}^{2}}+\frac{{{c}^{2}}}{{{t}^{2}}}={{a}^{2}}\] Þ \[{{c}^{2}}{{t}^{4}}-{{a}^{2}}{{t}^{2}}+{{c}^{2}}=0\]                    Product of roots will be, \[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=\frac{{{c}^{2}}}{{{c}^{2}}}=1\].

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