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find the point on the line x+2/3 + y+1/2 +z-3/2  at a distance 5 units from point (1,2,3).

Asked by upmaniousagarika 31st May 2016, 5:53 PM
Answered by Expert
Answer:
C o n s i d e r space t h e space g i v e n space e q u a t i o n comma
fraction numerator x plus 2 over denominator 3 end fraction equals fraction numerator y plus 1 over denominator 2 end fraction equals fraction numerator z minus 3 over denominator 2 end fraction equals k comma space w h e r e space k space i s space a n y space c o n s t a n t
therefore A n y space p o i n t space o n space t h e space l i n e space i s space o f space t h e space f o r m comma space P open parentheses 3 k minus 2 comma space 2 k minus 1 comma space 2 k plus 3 close parentheses
C o n s i d e r space t h e space p o i n t space Q open parentheses 1 comma 2 comma 3 close parentheses
G i v e n space t h a t space P Q space i s space 5 space u n i t s
T h u s comma
P Q squared equals open parentheses 3 k minus 2 minus 1 close parentheses squared plus open parentheses 2 k minus 1 minus 2 close parentheses squared plus open parentheses 2 k plus 3 minus 3 close parentheses squared
rightwards double arrow 5 squared equals open parentheses 3 k minus 2 minus 1 close parentheses squared plus open parentheses 2 k minus 1 minus 2 close parentheses squared plus open parentheses 2 k plus 3 minus 3 close parentheses squared
rightwards double arrow 25 equals open parentheses 3 k minus 3 close parentheses squared plus open parentheses 2 k minus 3 close parentheses squared plus open parentheses 2 k close parentheses squared
rightwards double arrow 25 equals 9 k squared plus 9 minus 18 k plus 4 k squared plus 9 minus 12 k plus 4 k squared
rightwards double arrow 25 equals 17 k squared minus 30 k plus 18
rightwards double arrow 17 k squared minus 30 k minus 7 equals 0
rightwards double arrow k equals fraction numerator 30 plus-or-minus square root of 30 squared minus 4 cross times 17 cross times open parentheses negative 7 close parentheses end root over denominator 2 cross times 17 end fraction
rightwards double arrow k equals fraction numerator 30 plus-or-minus square root of 1376 over denominator 34 end fraction
rightwards double arrow k equals fraction numerator 30 plus-or-minus 4 square root of 86 over denominator 34 end fraction
rightwards double arrow k equals fraction numerator 15 plus-or-minus 2 square root of 86 over denominator 17 end fraction
S u b s i t u t i n g space t h e space v a l u e space o f space k space i n space t h e space c o o r d i n a t e s space o f space t h e space p o i n t space P space w e space h a v e comma
P open parentheses 3 cross times fraction numerator 15 plus-or-minus 2 square root of 86 over denominator 17 end fraction minus 2 comma space 2 cross times fraction numerator 15 plus-or-minus 2 square root of 86 over denominator 17 end fraction minus 1 comma space 2 cross times fraction numerator 15 plus-or-minus 2 square root of 86 over denominator 17 end fraction plus 3 close parentheses
O n space s i m p l i f i c a t i o n comma space k equals 1.9734 space o r space k equals negative 0.2087
T h u s comma space t h e space p o i n t space P space b e c o m e space P open parentheses 3.9202 comma 2.9468 comma 6.9468 close parentheses space o r space P open parentheses negative 2.6261 comma space minus 1.4174 comma space 2.5826 close parentheses
Answered by Expert 1st June 2016, 12:51 PM
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