KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    In a \[\Delta ABC\] bisector of \[\angle C\] meets the side AB at D and circumcentre at E. The maximum value of \[CD\cdot DE\] is equal to

    A) \[\frac{{{a}^{2}}}{4}\]                                   

    B) \[\frac{{{b}^{2}}}{4}\]

    C) \[\frac{{{c}^{2}}}{4}\]           

    D) \[\frac{{{(a+b)}^{2}}}{4}\]

    Correct Answer: C

    Solution :

    \[CD\times DE=AD\times BD\]
    \[\frac{AD+DB}{2}\ge \sqrt{AD\times BD}\]                     \[[\because \,\,\,\,\,AB\ge 4M]\]
    \[\frac{c}{2}\ge \sqrt{CD\times DE}\]                   \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,CD\times DE\le \frac{{{c}^{2}}}{4}\]

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