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Three Dimensional Geometry

Short Answer Type

91. Find the length of perpendicular from (2, 1, 3) on the line

fraction numerator straight x minus 4 over denominator 5 end fraction space equals fraction numerator straight y minus 2 over denominator 4 end fraction space equals space fraction numerator straight z minus 3 over denominator 3 end fraction.

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92.
Find the length and foot of the perpendicular drawn from the point (3, 4, 5) on the line
$fraction numerator straight x minus 2 over denominator 2 end fraction space equals space fraction numerator straight y minus 3 over denominator 5 end fraction space equals space fraction numerator straight z minus 1 over denominator 3 end fraction$

Let the given line AB be

fraction numerator straight x minus 2 over denominator 2 end fraction space equals space fraction numerator straight y minus 3 over denominator 5 end fraction space equals space fraction numerator straight z minus 1 over denominator 3 end fraction

Any point M on this line is
(2 r + 2, 5 r + 3, 3 r + 1)
Let this point M be the foot of perpendicular from P (3, 4, 5) on AB
Direction-ratios of PM are
2 r + 2 – 3, 5 r + 3 – 4, 3 r + 1 – 5
i.e. 2 r – 1, 5 r – 1, 3 – 4
Direction-ratios of AB are 2, 5, 3
Since PM ⊥ AB
$therefore space space space space left parenthesis 2 straight r minus 1 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 5 straight r minus 1 right parenthesis thin space left parenthesis 5 right parenthesis space plus space left parenthesis 3 straight r minus 4 right parenthesis thin space left parenthesis 3 right parenthesis space equals space 0 therefore space space space space 4 straight r minus 2 plus 25 straight r minus 5 plus 9 straight r minus 12 space equals space 0 space space space space space space rightwards double arrow space space 38 space straight r space space equals space 19 therefore space space space space space space space space space space space space straight r space equals space 1 half therefore space space space space foot space of space perpendicular space straight M space is space open parentheses 1 plus 2 comma space 5 over 2 plus 3 comma space 3 over 2 plus 1 close parentheses space space space straight i. straight e. space space space space space open parentheses 3 comma space 11 over 2 comma space 5 over 2 close parentheses$

Required length of perpendicular = PM

$equals space square root of left parenthesis 3 minus 3 right parenthesis squared plus open parentheses 4 minus 11 over 2 close parentheses squared plus open parentheses 5 minus 5 over 2 close parentheses squared end root space equals space square root of 0 plus 9 over 4 plus 25 over 4 end root space equals space square root of 34 over 4 end root space equals space fraction numerator square root of 34 over denominator 2 end fraction.$

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93.

Find the perpendicular distance of the point (1, 0, 0) form the line $fraction numerator straight x minus 1 over denominator 2 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 3 end fraction space equals space fraction numerator straight z plus 10 over denominator 8 end fraction$

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94.

Find the length of the perpendicular drawn from the point (1, ,2 3) on the line
$fraction numerator straight x minus 6 over denominator 3 end fraction space equals space fraction numerator straight y minus 7 over denominator 2 end fraction space equals space fraction numerator straight z minus 7 over denominator negative 2 end fraction$