KVPY Sample Paper KVPY Stream-SX Model Paper-22

  • question_answer
    A chord of the parabola \[y=-\,{{a}^{2}}{{x}^{2}}+5ax-4\] touches the curve \[y=\frac{1}{1-x}\] at the point \[x=2\] and is bisected by that point. If S is the sum of all possible values of a, then find 12S.

    A) 12

    B) 15

    C) 17

    D) 19

    Correct Answer: A

    Solution :

    Slope of the tangent \[{{\left. \frac{dy}{dx}=\frac{1}{{{(1-x)}^{2}}}\, \right|}_{x\,=\,2}}=1\]
    Equation of tangent is \[y+1=1\,(x-2)\]
    i.e.        \[y=x-3\]
    parabola \[y=-{{a}^{2}}{{x}^{2}}+5ax-4\]
    Solving the equations of tangent and the parabola
    Since x is real     \[\therefore \]      \[{{(1-5a)}^{2}}-4{{a}^{2}}\ge 0\]
    \[\Rightarrow \]\[a\le \frac{1}{7}\] or \[a\ge \frac{1}{3}\]
    Sum of roots  \[=\frac{5a-1}{{{a}^{2}}}=2\times 2\]
    \[a=1,\]\[\frac{1}{4}\]                [\[a=\frac{1}{4}\]is rejected]
    \[\therefore \]      \[S=1\]              \[\therefore \]\[12S=12\]

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