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CBSE Class 12-science Answered

Cartesian equation 
question image
Asked by alanpeter9611 | 24 Feb, 2019, 10:58: PM
answered-by-expert Expert Answer
Equation of plane is given by
 
begin mathsize 12px style open parentheses stack space r space with rightwards arrow on top minus space stack space a space with rightwards arrow on top space close parentheses. space stack space n space with rightwards arrow on top space comma space w h e r e space space stack space r space with rightwards arrow on top space i s space p o s i t i o n space v e c t o r space o f space a space p o i n t space i n space t h e space p l a n e comma space stack space a space with rightwards arrow on top space p o s i t i o n space v e c t o r space o f space a space g i v e n space p o i n t space i n space t h e space p l a n e space a n d space stack space n space with rightwards arrow on top space i s space t h e space n o r m a l space t o space p l a n e end style
 
Line segment joining the points A (2,3,5)  and  B(4,5,7) is begin mathsize 12px style fraction numerator x space minus 2 over denominator 4 minus 2 end fraction space equals space fraction numerator y minus 3 over denominator 5 space minus space 3 end fraction space equals space fraction numerator z minus 5 over denominator 7 minus 5 end fraction space space space space space space o r space space space space space space space fraction numerator begin display style x space minus 2 end style over denominator begin display style 2 end style end fraction space equals space fraction numerator begin display style y minus 3 end style over denominator begin display style 2 end style end fraction space equals space fraction numerator begin display style z minus 5 end style over denominator begin display style 2 end style end fraction space end style ...............(1)
mid point M between A and B  = M (3,4,6)
 
M(3,4,6) is on the plane, hence begin mathsize 12px style stack space a with rightwards arrow on top space equals space 3 space i with hat on top space plus space 4 space j with hat on top space plus space 6 space k with hat on top space end style
line AB is normal to plane and direction ratio of AB can be considered as 1,1 and 1 .  
 
Hence begin mathsize 12px style stack space n with rightwards arrow on top space equals space i with hat on top space plus space j with hat on top space plus space k with hat on top space end style
Hence equation of plane is  begin mathsize 12px style open parentheses stack space r space with rightwards arrow on top space minus space open parentheses 3 space i with hat on top space plus space 4 space j with hat on top space plus space 6 space k with hat on top close parentheses space close parentheses times open parentheses i with hat on top space plus space j with hat on top space plus space k with hat on top close parentheses space end style
begin mathsize 12px style sin c e space r with rightwards arrow space on top equals space open parentheses x space i with hat on top space plus space y space j with hat on top space plus space z space k with hat on top close parentheses e q u a t i o n space o f space p l a n e space i s space space open parentheses x space i with hat on top space plus space y space j with hat on top space plus space z space k with hat on top close parentheses. space open parentheses i with hat on top space plus space j with hat on top space plus k with hat on top close parentheses space equals space open parentheses 3 space i with hat on top space plus space 4 space j with hat on top space plus space 6 space k with hat on top close parentheses. open parentheses i with hat on top space plus space j with hat on top space plus k with hat on top close parentheses x space plus y plus z space space equals space 13 end style
Answered by Thiyagarajan K | 25 Feb, 2019, 01:28: AM

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