Question

If the lines $\frac{x-1}{2}=\frac{y+3}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then the value of $k$ is

• A. $\frac{3}{2}$
• B. $\frac{9}{2}$
• C. $-\frac{2}{9}$
• D. $-\frac{3}{2}$

Answer 1

Answer: (B)

Since, the lines intersect, therefore they must have a point in common, i.e.
$ \frac{x-1}{2} = \frac{y+1}{3}=\frac{z-1}{4} = \lambda $
and $ \frac{x-3}{1} = \frac{y-k}{2}=\frac{z}{1} = \mu $
$\Rightarrow x=2 \lambda + 1,y = 3 \lambda -1$
$ z = 4 \lambda + 1$
and $\, \, \, x = \mu +3, y = 2\, \mu + k,z = \mu$ are same.
$\Rightarrow 2 \lambda + 1 = \mu +3$
$ 3 \lambda - 1 = 2\mu + k$
$ 4 \lambda + 1 = \mu $
On solving Ist and IIIrd terms, we get, $ \lambda = - \frac{3}{2} $
and $ \mu = -5$
$\therefore k =3 \lambda - 2\mu - 1$
$ \Rightarrow k =3 \bigg(-\frac{3}{2}\bigg)-2(-5)-1 =\frac{9}{2}$
$ \therefore k =\frac{9}{2}$