Junior Class 1 Answered

Find the Cartesian equation of the line passing through the origin and parallel to the line of intersection of the planes x+3y+6=0 and 3x-y-4z=0.

Asked by Arun Vigesh | 08 Feb, 2017, 06:27: PM

Expert Answer

$begin mathsize 14px style Let space the space direction space ratios space of space the space required space line space be space proportional space to space straight a comma space straight b comma space straight c. So space the space equation space that space is comma fraction numerator straight x minus 0 over denominator straight a end fraction equals fraction numerator straight y minus 0 over denominator straight b end fraction equals fraction numerator straight z minus 0 over denominator straight c end fraction rightwards double arrow straight x over straight a equals straight y over straight b equals straight z over straight c space.... left parenthesis straight i right parenthesis Since space left parenthesis straight i right parenthesis space is space parallel space to space the space line space of space intersection space of space the space planes space straight x plus 3 straight y plus 6 equals 0 space and space 3 straight x minus straight y minus 4 straight z equals 0 rightwards double arrow space space space space straight a plus 3 straight b equals 0 space and space 3 straight a minus straight b minus 4 straight c equals 0 Using space cross space multiplication comma space we space get fraction numerator straight a over denominator left parenthesis 3 right parenthesis left parenthesis negative 4 right parenthesis minus left parenthesis 0 right parenthesis left parenthesis negative 1 right parenthesis end fraction equals fraction numerator straight b over denominator left parenthesis 0 right parenthesis left parenthesis 3 right parenthesis minus left parenthesis 1 right parenthesis left parenthesis negative 4 right parenthesis end fraction equals fraction numerator straight c over denominator left parenthesis 1 right parenthesis left parenthesis negative 1 right parenthesis minus left parenthesis 3 right parenthesis left parenthesis 3 right parenthesis end fraction rightwards double arrow fraction numerator straight a over denominator negative 12 end fraction equals straight b over 4 equals fraction numerator straight c over denominator negative 10 end fraction rightwards double arrow fraction numerator straight a over denominator negative 6 end fraction equals straight b over 2 equals fraction numerator straight c over denominator negative 5 end fraction equals straight r space left parenthesis say right parenthesis rightwards double arrow straight a equals negative 6 straight r comma space space space space space straight b equals 2 straight r comma space space space space straight c equals negative 5 straight r Substituting space the space values space of space straight a comma space straight b comma space and space straight c space in space left parenthesis straight i right parenthesis comma space we space obtain space the space equation space of space the space required space line space as fraction numerator straight x over denominator negative 6 end fraction equals straight y over 2 equals fraction numerator straight z over denominator negative 5 end fraction end style$