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Integrals

Question
CBSEENMA12035645
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Find the coordinates of the foot of perpendicular drawn from the point A
(-1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1). Hence find the image of the point A in the line BC.

Solution

Suppose P be the foot of the perpendicular drawn from point A on the line joining
points B and C.
Let P’ (a, b, c) be the coordinates of the image of point A.
fraction numerator straight x minus straight x subscript 1 over denominator straight x subscript 2 minus straight x subscript 1 end fraction space equals space fraction numerator straight y minus straight y subscript 1 over denominator straight y subscript 2 minus end subscript straight y subscript 1 end fraction space space equals space fraction numerator straight z minus straight z subscript 1 over denominator straight z subscript 2 minus straight z subscript 1 end fraction fraction numerator straight x minus 0 over denominator 2 end fraction space equals space fraction numerator space straight y plus 1 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 3 over denominator negative 4 end fraction space equals space straight lambda General space coordinates space of space straight P space is space left parenthesis 2 straight lambda comma negative 2 straight lambda minus 1 comma negative 4 straight lambda plus 3 right parenthesis Direction space ratio space of space AP space left parenthesis 2 straight lambda plus 1 comma negative 2 straight lambda minus 9 comma 4 straight lambda minus 1 right parenthesis AP perpendicular BC 2 left parenthesis 2 straight lambda plus 1 right parenthesis minus 2 left parenthesis negative 2 straight lambda minus 9 right parenthesis minus 4 left parenthesis negative 4 straight lambda minus 1 right parenthesis equals 0 4 straight lambda plus 2 plus 4 straight lambda plus 18 plus 16 straight lambda plus 4 space equals 0 24 plus 24 straight lambda space equals 0 straight lambda equals negative 1 straight P left parenthesis negative 2 comma 1 comma 7 right parenthesis Coordinates space of space foot space of space perpendicular space is space left parenthesis negative 2 comma 1 comma 7 right parenthesis Coordinates space of space image space of space straight A space is space straight P to the power of apostrophe left parenthesis space straight a comma straight b comma straight c right parenthesis space is space fraction numerator straight a minus 1 over denominator 2 end fraction equals negative 2 comma space straight a equals negative 3 fraction numerator straight b plus 8 over denominator 2 end fraction space equals space 1 comma space straight b equals negative 6 fraction numerator straight c plus 4 over denominator 2 end fraction space equals space 7 comma space straight c equals space 10 straight P apostrophe space left parenthesis negative 3 comma negative 6 comma 10 right parenthesis

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