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Consider two planes P_(1):2x-3y++z=-5 and P_(2):x-4y-14z=5 if a line L whose points A-=(-7,y_(1),z_(1)) lie on plate P_(1) and points B-=(X_(2),y_(2),z_(2)) and C-=(x_(3),y_(3),z_(3)) lies on plate P_(2) has equation (x+7)/(a)=(y-y_(1))/(-b)=(z-z_(1))/(c) where a,b,c epsilonN and a+b+c is minimum. Q. The minimum value of a+b+c is

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Solution :Line `L:(x+7)/(a)=(y-y_(1))/(-b)=(z-z_(1))/(c)`
`(-7,y,z_(1))` lie on plane `P_(1)`
`implies-3y_(1)+z_(1)=9` ,…(1)
Line L LIES on plate `P_(2)`
`impliesa+4b+14c=0`..(2)
Also `(-7,y_(1),z_(1))` lies on `P_(2)`
`implies-4y_(1)-14z_(1)=12` .. (3)
Form (1) and (3)
Line passes through `(-7,-3,0)`
Least value of a,b,c is `a=2,b=3,c=1`
(from (2))


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