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    Question

    The shortest distance between the two straight linefraction numerator x minus 4 divided by 3 over denominator 2 end fraction equals fraction numerator y plus 6 divided by 5 over denominator 3 end fraction equals fraction numerator z minus 3 divided by 2 over denominator 4 end fraction and fraction numerator 5 y plus 6 over denominator 8 end fraction equals fraction numerator 2 z minus 3 over denominator 9 end fraction equals fraction numerator 3 x minus 4 over denominator 5 end fraction is

    1. square root of 29
    2. 3
    3. 0
    4. 6 square root of 10

    Hint:

    In the question two lines are given in the cartesian formfraction numerator x minus x subscript 1 over denominator a subscript 1 end fraction equals fraction numerator y minus y subscript 1 over denominator b subscript 1 end fraction equals fraction numerator z minus z subscript 1 over denominator c subscript 1 end fraction space a n d space fraction numerator x minus x subscript 2 over denominator a subscript 2 end fraction equals fraction numerator y minus y subscript 2 over denominator b subscript 2 end fraction equals fraction numerator z minus z subscript 2 over denominator c subscript 2 end fraction we have to find the shortest distance between the two lines with the formula fraction numerator 1 over denominator square root of D end fraction open vertical bar table row cell x subscript 2 minus x subscript 1 end cell cell y subscript 2 minus y subscript 1 end cell cell z subscript 2 minus z subscript 1 end cell row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell a subscript 2 end cell cell b subscript 2 end cell cell c subscript 2 end cell end table close vertical bar space comma w h e r e space D equals left parenthesis a subscript 1 b subscript 2 minus a subscript 2 b subscript 1 right parenthesis squared plus left parenthesis b subscript 1 c subscript 2 minus b subscript 2 c subscript 1 right parenthesis squared plus left parenthesis c subscript 1 a subscript 2 minus c subscript 2 a subscript 1 right parenthesis squared.

    The correct answer is: 0

      The given equations are
      fraction numerator x minus 4 divided by 3 over denominator 2 end fraction equals fraction numerator y plus 6 divided by 5 over denominator 3 end fraction equals fraction numerator z minus 3 divided by 2 over denominator 4 end fraction and
      fraction numerator 5 y plus 6 over denominator 8 end fraction equals fraction numerator 2 z minus 3 over denominator 9 end fraction equals fraction numerator 3 x minus 4 over denominator 5 end fraction
      rightwards double arrow fraction numerator x minus begin display style 4 over 3 end style over denominator begin display style 5 over 3 end style end fraction equals fraction numerator y plus begin display style 6 over 5 end style over denominator begin display style 8 over 5 end style end fraction equals fraction numerator z minus begin display style 3 over 2 end style over denominator begin display style 9 over 2 end style end fraction
      The shortest distance between the two cartesian lines
      =fraction numerator 1 over denominator square root of D end fraction open vertical bar table row cell x subscript 2 minus x subscript 1 end cell cell y subscript 2 minus y subscript 1 end cell cell z subscript 2 minus z subscript 1 end cell row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell a subscript 2 end cell cell b subscript 2 end cell cell c subscript 2 end cell end table close vertical bar space comma w h e r e space D equals left parenthesis a subscript 1 b subscript 2 minus a subscript 2 b subscript 1 right parenthesis squared plus left parenthesis b subscript 1 c subscript 2 minus b subscript 2 c subscript 1 right parenthesis squared plus left parenthesis c subscript 1 a subscript 2 minus c subscript 2 a subscript 1 right parenthesis squared
      equals fraction numerator 1 over denominator square root of D end fraction open vertical bar table row cell 4 over 3 minus 4 over 3 end cell cell fraction numerator negative 6 over denominator 5 end fraction plus 6 over 5 end cell cell 3 over 2 minus 3 over 2 end cell row 2 3 4 row cell 5 over 3 end cell cell 8 over 5 end cell cell 9 over 2 end cell end table close vertical bar equals fraction numerator 1 over denominator square root of D end fraction open vertical bar table row 0 0 0 row 2 3 4 row cell 5 over 3 end cell cell 8 over 5 end cell cell 9 over 2 end cell end table close vertical bar equals fraction numerator 1 over denominator square root of D end fraction cross times 0 equals 0

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