The equation of the plane passing through the points (1,2,3), (-1,4,2) and (3,1,1) is

Options

A. `5x+y+12z-23=0`
B. `5x+6y+2z-23=0`
C. `x+6y+2z-13=0`
D. `x+y+z-13=0`

Answer

B

Explanation

Equation of plane through (1,2,3) is
`a(x-1)+b(y-2)+c(z-3)=0.....(i)`
`:.` It passes through `(-1,,4,2) and (3,1,1)`
`:. -2a +2b-c=0 and 2a-b=2c=0`
`rArr (a)/(-5)=(b)/(-6)=(c)/(-2)`
`:.` Equation of plane is
`-5x-6y-2z+5+12+6=0`
`rArr 5x+6y+2z-23=0`
Alternate solution
Equation solution
`|{:(x-x_(1),y-y_(1),z-z_(1)),(x_(2)-x_(1),y_(2)-y_(2),z_(2)-z_(1)),(x_(3)-x_(1),y_(3)-y_(1),z_(3)-z_(1)):}|=0`
`rArr |{:(x-1,y-2,z-3),(-2,2,-1),(2,-1,-2):}|=0`
`rArr (x-1)(-4-1)-(y-2)(4+2)+(z-3)(2-4)=0`
`rArr= -5x+5-6y+12-2z+6=0`
`rArr 5x+6y+2z-23=0`