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JEE Main & Advanced Sample Paper JEE Main Sample Paper-45

  • question_answer
    Let a variable line has its intercepts on the coordinate axes, respectively as e,e' where \[\frac{e}{2}\]and \[\frac{e'}{2}\] are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}},\] where a is equal to

    A)  4                                

    B)  3              

    C)  2                                

    D)  1

    Correct Answer: C

    Solution :

     Since, \[\frac{e}{2}\] and \[\frac{e'}{2}\] are the eccentricities of a hyperbola and its conjugate hyperbola. Then, \[\frac{1}{{{\left( \frac{e}{2} \right)}^{2}}}+\frac{1}{{{\left( \frac{e'}{2} \right)}^{2}}}=1\] \[\Rightarrow \]            \[1=\frac{4}{{{e}^{2}}}+\frac{4}{e{{'}^{2}}}\] \[\Rightarrow \]            \[4=\frac{{{e}^{2}}e{{'}^{2}}}{{{e}^{2}}+{{e}^{'2}}}\]             ?(i)


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