Question

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ 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

The correct option is B

$\frac{9}{2}$

Since the lines intersect, therefore they must have a point in common, i.e.
$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}=\lambda and\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}=\mu \Rightarrow x=2\lambda +1,y=3\lambda -1z=4\lambda +1$
and $x=\mu +3,y=2\mu +k,z=\mu$ are same.
$\Rightarrow 2\lambda +1=\mu +33\lambda -1=2\mu +k4\lambda +1=\mu$
On solving these equations, we get,
$\lambda =-\frac{3}{2}and\mu =-5\therefore k=3\lambda -2\mu -1\Rightarrow k=3(-\frac{3}{2})-2(-5)-1=\frac{9}{2}\therefore k=\frac{9}{2}$