If lines `(x-1)/(2)=(y+1)/(3)=(z-1)/(4) and x-3=(y-k)/(2)=z` intersect, then the value of k is

Options

A. `(9)/(2)`
B. `(1)/(2)`
C. `(5)/(2)`
D. `(7)/(2)`

Answer

A

Explanation

The given lines are intersecting, since directing ratios of lines are not proportional.
Let `(x-1)/(2)=(y+1)/(3)=(z-1)/(4)=lamda`
`therefore`The coordinates of a point on the line are
`(2lamda,1,3lamda-1,4lamda+1)`
Similarly, let `(x-3)/(1)=(y-k)/(2)=(z-0)/(1)=mu`
`therefore` The coordinates of a point on the line are
`(mu+3,2mu+k,mu)`
Since, two lines intersect for some values of `lamda` and `mu`.
`therefore(2lamda+1,3lamda-1,4lamda+1)=(mu+3,2mu+k,mu)`
On comparing corresponding coordinates, we get
`2lamda+1=mu+3,3lamda-1=2mu+k and 4lamda+1=mu`
`implies2lamda-mu=2,3lamda-2mu=k+1 and 4lamda-mu=-1`
now, `4lamda-mu-2lamda+mu=-1-2`
`implies2lamda=-3`
`implieslamda=(-3)/(2)`
On putting `lamda=(-3)/(2)` in `2lamda-mu=2`, we get
`2xx((-3)/(2))-mu=2implies-3-mu=2impliesmu=-3-2=-5`
On putting `lamda=(-3)/(2) and mu=-5` in `3lamda-2mu=k+1`, we get
`3xx((-3)/(2))-2xx-5=k+1implies(-9)/(2)+10-1=k`
`impliesk=(-9+18)/(2)=(9)/(2)`