KVPY Sample Paper KVPY Stream-SX Model Paper-22

  • question_answer
    The line \[2px\,+\,y\,\sqrt{1-{{p}^{2}}}\,=\,1\], \[(\,\left| p \right|\,<\,1\,)\] for different values of p, touches

    A) an ellipse of eccentricity \[\sqrt{3}\,/\,2\]          

    B) an ellipse of eccentricity \[1\,/\sqrt{3}\,\]

    C) a hyperbola of eccentricity 2             

    D) an ellipse or a hyperbola depending on p

    Correct Answer: A

    Solution :

    i.e.        \[y=-\frac{2p}{\sqrt{1-{{p}^{2}}}}x+\frac{1}{\sqrt{1-{{p}^{2}}}}\] ?(i)
    Let \[m=-\frac{2p}{\sqrt{1-{{p}^{2}}}}.\] Then \[1-{{p}^{2}}=\frac{4}{4+{{m}^{2}}}.\]
    \[\therefore \]      equation of the line (i) becomes
    i.e.        \[y=mx+\sqrt{\frac{{{m}^{2}}}{4}+1}\]
    \[\therefore \]      the curve is an ellipse for which
                \[{{b}^{2}}=\frac{1}{4}\] and \[{{a}^{2}}=1\]
    \[\therefore \]      \[{{e}^{2}}=\frac{3}{4}\]                    \[\therefore \]      \[e=\frac{\sqrt{3}}{2}\]

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