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Home / Doubts and Solutions/CBSE/Class 12 Humanities/Mathematics

find equation of plane through the point (4,-3,2) and perpendicular to line of intersection ofplanes x-y+2z=3 and 2x-y-3z=0

Asked by MANMOHAN 2nd February 2017, 6:41 PM
Answered by Expert
Answer:

begin mathsize 16px style Given space that space straight x minus straight y plus 2 straight x minus 3 equals 0 rightwards double arrow straight x space plus space 2 straight z minus 3 equals straight y space... left parenthesis straight i right parenthesis 2 straight x minus straight y minus 3 straight z equals 0 rightwards double arrow 2 straight x minus 3 straight z equals straight y space.... left parenthesis ii right parenthesis Multiplying space left parenthesis straight i right parenthesis space by space 2 comma space we space get 4 straight x minus 2 straight y minus 6 straight z equals 0 Subtracting space left parenthesis ii right parenthesis space from space left parenthesis straight i right parenthesis comma space we space get straight y equals 7 straight z minus 6 rightwards double arrow straight z equals fraction numerator straight y plus 6 over denominator 7 end fraction Substituting space in space left parenthesis ii right parenthesis comma space we space get 2 straight x minus 3 open parentheses fraction numerator straight y plus 6 over denominator 7 end fraction close parentheses equals straight y rightwards double arrow 14 straight x minus 3 straight y minus 18 equals 7 straight y rightwards double arrow straight y equals fraction numerator 7 straight x minus 9 over denominator 5 end fraction So comma space the space line space of space intersection space is comma fraction numerator 7 straight x minus 9 over denominator 5 end fraction equals 7 straight z minus 6 So comma space the space direction space ratio space vector space is space 5 straight i with hat on top plus straight j with hat on top plus straight k with hat on top As space plane space is space perpendicular space to space line comma space hence space the space direction space ratios space of space line space will space be space perpendicular space to space direction space normal space of space plane. rightwards double arrow straight a straight i with hat on top plus straight b straight j with hat on top plus straight c straight k with hat on top equals 5 straight i with hat on top plus straight j with hat on top plus straight k with hat on top So comma space comparing space we space get comma straight a equals 5 comma space straight b equals 1 comma space straight c equals 1 ax plus by plus cz plus straight d equals 0 rightwards double arrow 5 straight x plus straight y plus straight z plus straight d equals 0 Since space it space passes space through space left parenthesis 4 comma space minus 3 comma space 2 right parenthesis rightwards double arrow 20 minus 3 plus 2 plus straight d equals 0 rightwards double arrow straight d equals negative 19 Hence comma space the space equaion space of space the space plane space is space 5 straight x plus straight y plus straight z minus 19 equals 0. end style

Answered by Expert 3rd February 2017, 5:39 PM
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