Question

If the angle between the line x = $fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 3 over denominator straight lambda end fraction$ and the plane x + 2y + 3z = 4 is cos-1 $square root of 5 over 14 end root$ then λ equal

2/3

3/2

2/5

3/5

Solution

2/3

fraction numerator straight x minus 0 over denominator 1 end fraction space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 3 over denominator straight lambda end fraction space... space left parenthesis 1 right parenthesis

x + 2y + 3z = 4 ....... (2)

Angle between the line and plane is
$cos space left parenthesis 90 minus space straight theta right parenthesis space equals space fraction numerator straight a subscript 1 straight a subscript 2 space plus straight b subscript 1 straight b subscript 2 space plus space straight c subscript 1 straight c subscript 2 over denominator square root of straight a subscript 1 superscript 2 plus straight b subscript 1 superscript 2 plus straight c subscript 1 superscript 2 end root space square root of straight a subscript 2 superscript 2 plus straight b subscript 2 superscript 2 plus straight c subscript 2 superscript 2 end root end fraction rightwards double arrow space sin space straight theta space equals space fraction numerator 1 space plus space 4 space plus space 3 straight lambda over denominator square root of 14 space straight x space square root of 5 space plus space straight lambda squared end root end fraction space equals space fraction numerator 5 space plus space 3 straight lambda over denominator square root of 14 space straight x space end root square root of 5 space plus space straight lambda squared end root end fraction space... space left parenthesis 3 right parenthesis$
But given that angle between line and plane is

rightwards double arrow space sin space straight theta space equals space fraction numerator 3 over denominator square root of 14 end fraction therefore space from space left parenthesis 3 right parenthesis fraction numerator 3 over denominator square root of 14 end fraction space equals space fraction numerator space 5 space plus space 3 straight lambda over denominator square root of 14 space end root space straight x square root of space 5 space plus straight lambda squared end root end fraction rightwards double arrow space 9 space left parenthesis 5 space plus space straight lambda squared right parenthesis space equals space 25 space plus space 9 straight lambda squared space plus space 30 space straight lambda 30 space straight lambda space equals space 20 straight lambda space equals space 2 divided by 3

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