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JEE Main & Advanced Sample Paper JEE Main Sample Paper-24

  • question_answer
    The eccentricity of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] whose length of conjugate axis is equal to half of the distance between the focus is

    A)  \[\frac{4}{3}\]                                     

    B)  \[\frac{4}{\sqrt{3}}\]

    C)  \[\frac{2}{\sqrt{3}}\]                            

    D)  \[\sqrt{3}\]

    Correct Answer: C

    Solution :

    \[2b=\frac{1}{2}\,(2ae)\,=4{{b}^{2}}\,={{a}^{2}}{{e}^{2}}\] \[\Rightarrow \,4{{b}^{2}}\,={{a}^{2}}+{{b}^{2}}\,\Rightarrow \,3{{b}^{2}}={{a}^{2}}\] Hence, \[{{e}^{2}}=1+\frac{{{b}^{2}}}{{{a}^{2}}}\,=1+\frac{1}{3}\,=\frac{4}{3}\Rightarrow \,e=\frac{2}{\sqrt{3}}\]


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