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 :
$$\frac{9}{2}$$
Solution :
$$\eqalign{
& \frac{{x - 1}}{2} = \frac{{y + 1}}{3} = \frac{{z - 1}}{4} = \lambda \cr
& \Rightarrow x = 2\lambda + 1,\,\,y = 3\lambda - 1{\text{ and }}\,z = 4\lambda + 1 \cr
& \frac{{x - 3}}{1} = \frac{{y - k}}{2} = \frac{z}{1} = \mu \cr
& \Rightarrow x = 3 + \mu ,\,\,y = k + 2\mu \,\,{\text{and }}z = \mu \cr} $$
Since above lines intersect
$$\eqalign{
& \Rightarrow 2\lambda + 1 = 3 + \mu .....(1) \cr
& \,\,\,\,\,\,\,\,3\lambda - 1 = 2\mu + k.....(2) \cr
& \,\,\,\,\,\,\,\,\mu = 4\lambda + 1.....(3) \cr} $$
Solving (1) and (3) and putting the value of $$\lambda $$ and $$\mu $$ in (2) we get, $$k = \frac{9}{2}$$