Question

Statement 1: If `a` is an integer the the straight lines
`vecr=hati+2hati+3hatk+lamda(ahati+2hatj+3hatk)`
and `vecr=2hati+3hatj+hatk+mu(3hati+hatj+2hatk)` intersect at a point for `a=-5`.
Statement 2: Two straight lines intersect if the shortest distance between them is zero.
A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer 1

Correct Answer - A
Clearly, statement -2 is true.
Given lines pass through `veca_(1)=hati+2hatj+3hatk` and `veca_(2)=2hati+3hatj+3hatk`
respectively and are parallel to vectors `vecb_(1)=ahati+2hatj+3hatk` and `vecb_(2)=3hati+ahatj+2hatk` respectively.
If given lines intersect, then S.D `=0`
`implies(vecb_(1)xxvecb_(2)).(veca_(1)xxveca_(2))=0`
`implies|(a,2,3),(3,a,2),(1,1,-2)|=0`
`impliesa(-2a-2)-2(-6-2)+3(3-a)=0`
`implies2a^(2)+5a-25=0implies(a+5)(2a-5)=0implies=-5,5//2`