Sequences And Series

Question
CBSEENMA11015018

A (3, 2, 0), B (5, 3, 2) and C (-9, 6, -3) are the vertices of a triangle ABC. The bisector AD of ∆ BAC meets BC at D. Find the co-ordinates of D.

Solution

AD is bisector of angle BAC
rightwards double arrow  Ratio at D is c : b, where
             straight c space equals space AB space equals space square root of left parenthesis 3 minus 5 right parenthesis squared plus left parenthesis 2 minus 3 right parenthesis squared plus left parenthesis 0 minus 2 right parenthesis squared end root
                 equals space square root of 4 plus 1 plus 4 end root space equals space square root of 9 space equals space 3
             straight b space equals space AC space equals space square root of left parenthesis 3 plus 9 right parenthesis squared plus left parenthesis 2 minus 6 right parenthesis squared plus left parenthesis 0 plus 3 right parenthesis squared end root
             space space space equals space square root of 144 plus 16 plus 9 end root space equals space square root of 169 space equals space 13
                                                       
∴ For point D:
straight x space space equals space space fraction numerator straight c left parenthesis negative 9 right parenthesis plus straight b left parenthesis 5 right parenthesis over denominator straight c plus straight b end fraction equals fraction numerator 3 left parenthesis negative 9 right parenthesis plus 13 left parenthesis 5 right parenthesis over denominator 3 plus 13 end fraction equals 38 over 16 equals 19 over 8
straight y space space equals space space fraction numerator straight c left parenthesis 6 right parenthesis plus straight b left parenthesis 3 right parenthesis over denominator straight c plus straight b end fraction equals fraction numerator 3 left parenthesis 6 right parenthesis plus 13 left parenthesis 3 right parenthesis over denominator 3 plus 13 end fraction equals 57 over 16
straight z space space equals space space fraction numerator straight c left parenthesis negative 3 right parenthesis plus straight b left parenthesis 2 right parenthesis over denominator straight c plus straight b end fraction equals fraction numerator 3 left parenthesis negative 3 right parenthesis plus 13 left parenthesis 2 right parenthesis over denominator 3 plus 13 end fraction equals 17 over 16
Hence, point D is open parentheses 19 over 8 comma space 57 over 16 comma space 17 over 6 close parentheses