Question

A line from the origin meets the lines `(x-2)/1=(y-1)/-2=(z+1)/1` and `(x-8/3)/2=(y+3)/-1=(z-1)/1` at `P` and `Q` respectively. If length `PQ = d`, then `d^2` is
A. 3
B. 4
C. 5
D. 6

Answer 1

Correct Answer - D
Let the equation of a line passing through the originbe `x/a=y/b=z/c`. This meets the lines
`(x-2)/1=(y-1)/(-2)=(z+1)/(-1)` and `(x-8/3)/2=(y+3)/(-1)=(z-1)/1`
`:.|(2,1,-1),(1,-2,1),(a,b,c)|=0` and `|(8//3,-3,1),(2,-1,1),(a,b,c)|=0`
`impliesa+3b+5c=0` an `3a+b-5c=0`
`impliesa/5=b/(-5)=c/2`
Thus, the equation of the line through the origin intersection the given lines is
`x/5=y/(-5)=z/2`
The coordinates of any point on this line are `(54,-54,2r)`.
The coordinates of any point on `(x-2)/1=(y-1)/(-2)=(z+1)/1` are `(r_(1)+2,-2r_(1)+1,r_(1)-1)`.
If these two lines intersect then
`5r=r_(1)+2,-5r=-2r_(1)+1` and `2r=r_(1)-1`
`impliesr_(1)=3` and `r=1`
So the coordinates of `P` are `(5,-5,2)`.
Similarly coordinates of `Q` are `(10/3,(-10)/3,8/3)`
`:.PQ^(2)+(10/3-5)^(2)+((-10)/3+5)^(2)+(8/3-2)^(2)=6`