A (– 1, 2, – 3), B (5, 0, – 6), C (0, 4, – 1) are three points. Show that the direction cosines of the bisectors of are proportional to 25, 8, 5 and -11, $AB space equals space square root of left parenthesis 5 plus 1 right parenthesis squared plus left parenthesis 0 minus 2 right parenthesis squared plus left parenthesis negative 6 plus 3 right parenthesis squared end root space space space space space space equals space square root of 36 plus 4 plus 9 end root space equals space square root of 49 space equals space 7 AC space equals space square root of left parenthesis 0 plus 1 right parenthesis squared plus left parenthesis 4 minus 2 right parenthesis squared plus left parenthesis negative 1 plus 3 right parenthesis squared end root space space space space space space space space equals space square root of 1 plus 4 plus 4 end root space equals space square root of 9 space equals space 3$

A, B, C are the points (– 1, 2, – 3), (5, 0, – 6), (0, 4, – 1) respectively.

AB space equals space square root of left parenthesis 5 plus 1 right parenthesis squared plus left parenthesis 0 minus 2 right parenthesis squared plus left parenthesis negative 6 plus 3 right parenthesis squared end root space space space space space space space equals square root of 36 plus 4 plus 9 end root space equals space square root of 49 space equals space 7 AC space equals space square root of left parenthesis 0 plus 1 right parenthesis squared plus left parenthesis 4 minus 2 right parenthesis squared plus left parenthesis negative 1 plus 3 right parenthesis squared end root space space space space space space space equals square root of 1 plus 4 plus 4 end root space equals space square root of 9 space equals space 3

Syntax error from line 1 column 252 to line 1 column 355. Unexpected '<mlongdiv '.

because space space space BD over CD equals AB over AC equals 7 over 3

∴ D divides BC internally in the ratio 7 : 3 and E divides BC externally in the ratio 7 : 3

therefore space space space straight D space is space open parentheses fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis 0 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis 5 right parenthesis over denominator 7 plus 3 end fraction comma space fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis 0 right parenthesis over denominator 7 plus 3 end fraction comma space fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis negative 1 right parenthesis plus left parenthesis negative 3 right parenthesis thin space left parenthesis negative 6 right parenthesis over denominator 7 plus 3 end fraction close parentheses

i.e.,

open parentheses 15 over 10 comma space 28 over 10 comma space minus 25 over 10 close parentheses space space space straight i. straight e. comma space space space open parentheses 3 over 2 comma space 14 over 5 comma space minus 5 over 2 close parentheses

E is

open parentheses fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis 0 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis 5 right parenthesis over denominator 7 minus 3 end fraction comma space fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis 0 right parenthesis over denominator 7 minus 3 end fraction comma space fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis negative 6 right parenthesis over denominator 7 minus 3 end fraction close parentheses

i.e,

open parentheses negative 15 over 4 comma space 28 over 4 comma space 11 over 4 close parentheses space space space space straight i. straight e. comma space space space space open parentheses fraction numerator negative 15 over denominator 4 end fraction comma space 7 comma space 11 over 4 close parentheses

Directionratios of AD are

3 over 2 plus 1 comma space space 14 over 5 minus 2 comma space space fraction numerator negative 5 over denominator 2 end fraction plus 3

i.e.,

5 over 2 comma space space 4 over 5 comma space 1 half space space space or space space space 25 comma space 8 comma space 5

Direction ratios of AE are

negative 15 over 4 plus 1 comma space space 7 minus 2 comma space space 11 over 4 plus 3

i.e.,

negative 11 over 4 comma space 5 comma space 23 over 4 space space or space space space minus 11 comma space 20 comma space 33

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