4th sum

Asked by lovemaan5500 | 20th Jan, 2020, 09:31: PM

Expert Answer:

To find the coordinates of the points where the shortest distance between the two given lines meets them.
x-23/-6=y-19/-4=z-25/3   and     x-12/-9=y-1/4=z-5/2
 
 
T h e space l i n e s space a r e
fraction numerator x minus 23 over denominator negative 6 end fraction equals fraction numerator y minus 19 over denominator negative 4 end fraction equals fraction numerator z minus 25 over denominator 3 end fraction equals k space space... space left parenthesis i right parenthesis
fraction numerator x minus 12 over denominator negative 9 end fraction equals fraction numerator y minus 1 over denominator 4 end fraction equals fraction numerator z minus 5 over denominator 2 end fraction equals m space space... space left parenthesis i i right parenthesis
L e t space P space a n d space Q space b e space t h e space p o i n t s space l y i n g space o n space t h e space l i n e s space left parenthesis i right parenthesis space a n d space left parenthesis i i right parenthesis space r e s p e c t i v e l y.
T h e space c o o r d i n a t e s space o f space P space a n d space Q space w i l l space b e space o f space t h e space f o r m
P open parentheses negative 6 k plus 23 comma space minus 4 k plus 19 comma space 3 k plus 25 close parentheses space space space a n d space space space Q open parentheses negative 9 m plus 12 comma space 4 m plus 1 comma space 2 m plus 5 close parentheses
T h e r e f o r e comma space t h e space d i r e c t i o n space r a t i o s space o f space P Q space w i l l space b e space open parentheses negative 9 m plus 6 k minus 11 comma space 4 m plus 4 k minus 18 comma space 2 m minus 3 k minus 20 close parentheses
P Q space w i l l space b e space t h e space s h o r t e s t space d i s tan c e space i f space i t space i s space p e r p e n d i c u l a r space t o space b o t h space t h e space g i v e n space l i n e s
U s i n g space t h e space c o n d i t i o n space o f space p e r p e n d i c u l a r i t y comma space w e space h a v e
minus 6 open parentheses negative 9 m plus 6 k minus 11 close parentheses minus 4 open parentheses 4 m plus 4 k minus 18 close parentheses plus 3 open parentheses 2 m minus 3 k minus 20 close parentheses equals 0 space... space left parenthesis i i i right parenthesis
minus 9 open parentheses negative 9 m plus 6 k minus 11 close parentheses plus 4 open parentheses 4 m plus 4 k minus 18 close parentheses plus 2 open parentheses 2 m minus 3 k minus 20 close parentheses equals 0 space... space left parenthesis i v right parenthesis
S o l v i n g space left parenthesis i right parenthesis space a n d space left parenthesis i i right parenthesis comma space w e space h a v e
m equals 4 over 13 space a n d space k equals 95 over 91

Answered by Renu Varma | 21st Jan, 2020, 11:43: AM