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KVPY Sample Paper KVPY Stream-SX Model Paper-7

  • question_answer
    If the parabolas \[{{y}^{2}}=4b(x-c)\] and  \[{{y}^{2}}=8ax\] have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)?

    A) \[\left( \frac{1}{2},2,3 \right)\]

    B)  (1, 1, 3)

    C) \[\left( \frac{1}{2},2,0 \right)\]                 

    D) (1, 1, 0)

    Correct Answer: B

    Solution :

    Normal to there 2 curves are
    \[y=m(x-c)-2bm-b{{m}^{3}}\]
    \[y=mx-4am-2a{{m}^{3}}\]
    If they how a common normal
    \[(c+2b)-4a=(2a-b){{m}^{2}}\]
    (\[m=0\]corresponds to axis)
    \[\Rightarrow \]   \[{{m}^{2}}=\frac{c}{2a-b}-2>0\]
    \[\frac{c}{2a-b}>2\]
    According to the question answer is (1, 2, 3, 4) but may be in question they want common normal other than x-axis, hence answer is [B].


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