JEE Main & Advanced Mathematics Straight Line Question Bank Critical Thinking

  • question_answer Given vertices \[A(1,\,1),B(4,\,-2)\]and \[C(5,\,5)\]of a triangle, then the equation of the perpendicular dropped from C to the interior bisector of the angle A is                                              [Roorkee 1994]

    A)            \[y-5=0\]                                   

    B)            \[x-5=0\]

    C)            \[y+5=0\]                                  

    D)            \[x+5=0\]

    Correct Answer: B

    Solution :

               The internal bisector of the angle A will divide the opposite side \[BC\]at \[D\]in the ratio of arms of the angle i.e.\[AB=3\sqrt{2}\]and \[AC=4\sqrt{2}\]. Hence by ratio formula the point D is \[\left( \frac{31}{7},1 \right)\]. Slope of \[AD\]by \[\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=0\].                    \ Slope of a line perpendicular to \[AD\]is \[\infty \].                    Any line through C perpendicular to this bisector is \[\frac{y-5}{x-5}=m=\infty \]; \ \[x-5=0\].

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