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KVPY Sample Paper KVPY Stream-SX Model Paper-10

  • question_answer
    A point P lies on the hyperbola \[9{{x}^{2}}-16{{y}^{2}}=144\] such that \[P{{S}_{1}}:P{{S}_{2}}=3/2\] (where \[{{S}_{1}}\] and \[{{S}_{2}}\]are focii of hyperbola). Coordinates of point P is in the first quadrant are

    A) \[(4\sqrt{2},\,\,3)\]                     

    B) \[(8,\,\,3\sqrt{3})\]

    C) \[\left( 5,\,\,\frac{9}{4} \right)\]               

    D) \[(16,\,\,3\sqrt{15})\]

    Correct Answer: D

    Solution :

    \[\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{9}=1\] \[(4sec\theta ,\,\,3tan\theta )\]
    \[\frac{3}{2}=\frac{P{{S}_{1}}}{P{{S}_{2}}}=\frac{\left| \frac{4e\sec \theta +4}{e} \right|}{\left| \frac{4e\sec \theta -4}{e} \right|}=\left| \frac{e\sec \theta +1}{e\sec \theta -1} \right|\]
    \[\Rightarrow \]   \[\sec \theta =4\]
    and \[\tan \theta =\sqrt{16-1}=\sqrt{15}\]


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