If line joining points A and B having position vectors `6 bar a-4 bar b+4 bar c` and `-4 bar c` respectively, and the line joining the points C and D having position vectors `-bar a-2 bar b-3 bar c` and `bar a+2 bar b -5 bar c` intersect, then their point of intersection is:

Options

A. B
B. C
C. D
D. A

Answer

A

Explanation

Coordinate of points A and B are (6, -4, 4) and ( 0, 0, -4) and coordinate of points C and D are (-1, -2, -3) and (1, 2, -5). Now, equation of line passing through (0, 0, -4) and (6, -4,4) is
`(x-0)/(6)=(y-0)/(-4) =(z+4)/(4+4)=k" "["say"]`
`Rightarrow x=6k, y=-4k`
and `z=8k-4....(i)`
Again, equation of line passing through (-1,-2,-3) and (1,2,-5) is
`(x+1)/(1+1)=(y+2)/(2+2)=(z+3)/(-5+3)`
`Rightarrow (x+1)/(2)=(y+2)/(4)=(3+3)/(-2).....(ii)`
Since, two lines are intersect, therefore point (6k, -4k, Bk -4) satisfy Eq. (ii), we get
`(6k+1)/(2)=(-4k+2)/(4)=(8k-4+3)/(-2)`
`Rightarrow 6k+1=-2k+1=-(8k-1)`
`therefore 6k+1=-2k+1`
`Rightarrow 8k=0`
`Rightarrow k=0`
`therefore x=6 xx 0, y=-4 xx 0`
and `z=8xx0-4`
`Rightarrow x=0, y=0 and z=-4
which is equal to the B coordinate.