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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2008

  • question_answer
        If the distance between the foci and the distance between the directrices of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]are in the ratio\[3:2,\] then\[a:b\]is

    A)  \[\sqrt{2}:1\]                                   

    B)  \[\sqrt{3}:\sqrt{2}\]

    C)  \[1:2\]                                 

    D)  \[2:1\]

    Correct Answer: A

    Solution :

                    Equation of hyperbola is \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] Distance between foci \[=2ae\] and distance between directrices\[=\frac{2a}{e}\] According to the question,                 \[\frac{2ae}{2a/e}=\frac{3}{2}\] \[\Rightarrow \]               \[{{e}^{2}}=\frac{3}{2}\] We know,                 \[{{b}^{2}}={{a}^{2}}({{e}^{2}}-1)\] \[\Rightarrow \]               \[\frac{{{b}^{2}}}{{{a}^{2}}}=\frac{3}{2}-1=\frac{1}{2}\] \[\Rightarrow \]               \[\frac{b}{a}=\frac{1}{\sqrt{2}}\] \[\therefore \]Required ratio\[a:b\]is\[\sqrt{2}:1\].


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