KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    Two vertices of a triangle are \[(5,-1)\] and \[(-\,2,3)\] if orthocenter of the triangle is origin, then the coordinate of third vertex is

    A) \[(4,7)\]             

    B) \[(3,7)\]

    C) \[(-4,-7)\]           

    D) \[(4,-\,7)\]

    Correct Answer: C

    Solution :

    Slope of \[AH=\frac{k}{h}\]
    Slope of\[BC=\frac{4}{-\,7}\]
    Slope of \[AH\times \]slope of \[BC=-1\]
    \[\therefore \]      \[\frac{k}{h}\times \frac{4}{-\,7}=-1\]
    \[\frac{k}{h}=\frac{7}{4}\]                                                        ? (i)
    Slope of \[AC=\frac{k+1}{h-5}\]
    Slope of \[BH=\frac{-\,3}{2}\]
    Slope of \[AC\times \]Slope of \[BH=-\,1\]
    \[\therefore \]      \[\left( \frac{k+1}{h-5} \right)\left( \frac{-\,3}{2} \right)=-1\]
    \[\Rightarrow \]   \[\frac{k+1}{h-5}=\frac{2}{3}\]                         ? (ii)
    On solving Eqs. (i) and (ii), we get \[h=-\,4,k=-\,7\]
    \[\therefore \]Third vertex of triangle is \[(-\,4,-7).\]  

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