Question

Determine whether the following pair of lines intersect or not: 

$$\frac{x-1}{2}=\frac{y+1}{3}=z\text{ and }\frac{x+1}{5}=\frac{y-2}{1};z=2$$ 

Answer 1

$$\frac{x-1}{2}=\frac{y+1}{3}=z\text{ and }\frac{x+1}{5}=\frac{y-2}{1};z=2$$  

The coordinates of any point on the first line are given by 

$$\frac{x-1}{2}=\frac{y+1}{3}=z=\lambda$$

$$\Rightarrow x=2\lambda +1$$

$$y=3\lambda -1$$

$$z=\lambda$$

The coordinates of a general point on the first line are  $$(2\lambda +1,3\lambda -1,\lambda )$$ 

Also, the coordinates of any point on the second line are given by

$$\frac{x+1}{5}=\frac{y-2}{1}=\mu ,z=2$$

$$\Rightarrow x=5\mu -1$$

$$y=\mu +2$$

$$z=2$$

The coordinates of a general point on the second line are $$(5\mu -1,\mu +2,2)$$ 

If the lines intersect, then they have a common point. So, for some values of

$$\lambda \text{ and }\mu$$ ,

we must have ,

$$2\lambda +1=5\mu -1,3\lambda -1=\mu +2,\lambda =2$$

$$\Rightarrow 2\lambda -5\mu =-2...(1)$$

$$3\lambda -\mu =3...(2)$$

$$\lambda =2...(3)$$

$$\text{ Solving (2) and (3), we get }$$

$$\lambda =2$$

$$\mu =3$$

$$\text{ Substituting }\lambda =2\text{ and }\mu =3\text{ in}\left(1\right),\text{ we get }$$

$$LHS=2\lambda -5\mu$$

$$=2\left(2\right)-5\left(3\right)$$

$$=4-15$$

$$=-11\ne -2$$

$$\Rightarrow LHS\ne RHS$$

$$\text{ Since}\lambda =2\text{ and }\mu =3\text{ do not satisfy (1), the given lines do not intersect }.$$