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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 41

  • question_answer
    If a circles \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]and the rectangular hyperbola \[xy={{c}^{2}}\] intersect m four points, \[\left( c{{t}_{r}},\frac{c}{{{t}_{r}}} \right),\] \[r=1,2,3,4\]then \[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}\] is equal to

    A) \[-1\]                

    B)        \[1\]

    C) \[{{c}^{4}}\]              

    D)        \[-{{c}^{4}}\]

    Correct Answer: B

    Solution :

    Parametric equation of the hyperbola \[xy={{c}^{2}}\]is \[(ct,\,c/t)\] and equation of circle is \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]    ?..(i) Put \[x=ct\] and \[y=c/t\]in (i) \[{{(ct)}^{2}}+{{\left( \frac{c}{t} \right)}^{2}}={{a}^{2}}\] \[{{c}^{2}}{{t}^{4}}+{{c}^{2}}-{{a}^{2}}{{t}^{2}}=0\]            ?..(ii) From (ii), \[{{t}_{1}}\,{{t}_{2}}\,{{t}_{3}}\,{{t}_{4}}=\frac{{{c}^{2}}}{{{c}^{2}}}=1\]


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