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Linear Inequalities

Question
CBSEENMA11015478

The image of line fraction numerator straight x minus 1 over denominator 3 end fraction space equals space fraction numerator straight y minus 3 over denominator 1 end fraction space equals space fraction numerator straight z minus 4 over denominator negative 5 end fraction in the plane 2x-y+z+3 =0 is the line

  • fraction numerator straight x plus 3 over denominator 3 end fraction space equals space fraction numerator straight y minus 5 over denominator 1 end fraction space equals space fraction numerator straight z minus 2 over denominator negative 5 end fraction
  • fraction numerator straight x plus over denominator negative 3 end fraction space equals space fraction numerator straight y minus 5 over denominator negative 1 end fraction space equals space fraction numerator begin display style straight z plus 2 end style over denominator 5 end fraction
  • fraction numerator straight x minus 3 over denominator 3 end fraction space equals space fraction numerator straight y plus 5 over denominator 1 end fraction space equals space fraction numerator straight z minus 2 over denominator negative 5 end fraction
  • fraction numerator straight x minus 3 over denominator negative 3 end fraction space equals space fraction numerator straight y plus 5 over denominator negative 1 end fraction space equals fraction numerator straight z minus 2 over denominator 5 end fraction

Solution

A.

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

Here, plane and line are parallel to each other. Equation of normal to the plane through the point (1,3,4) is
fraction numerator straight x minus 1 over denominator 2 end fraction space equals space fraction numerator straight y minus 3 over denominator negative 1 end fraction space equals space fraction numerator straight z minus 4 over denominator 1 end fraction space equals space straight k
Any point in this normal is (2jk+2, -k+3,4+k)
Then, open parentheses fraction numerator 2 straight k plus 1 plus 1 over denominator 2 end fraction comma space fraction numerator 3 minus straight k plus 3 over denominator 2 end fraction comma space fraction numerator 4 plus straight k plus 4 over denominator 2 end fraction close parentheses lies on the plane.
rightwards double arrow space 2 left parenthesis straight k plus 1 right parenthesis space minus space open parentheses fraction numerator 6 minus straight k over denominator 2 end fraction close parentheses space plus open parentheses fraction numerator 8 plus straight k over denominator 2 end fraction close parentheses space plus 3 space equals 0
rightwards double arrow straight k space equals negative 2
hence, point through which this image pass is
(2k+1,3-k,4+k)
i.e., (2(-2) + 1, 3+2, 4-2) = (-3,5,2)
Hence equation of image line is
fraction numerator straight x plus 3 over denominator 3 end fraction space equals space fraction numerator straight y minus 5 over denominator 1 end fraction space equals space fraction numerator straight z minus 2 over denominator negative 5 end fraction

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