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  \[\left( 2\lambda + 1, 3\lambda - 1, \lambda \right)\] 

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 \[\left( 5\mu - 1, \mu + 2, 2 \right)\] 

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

\[\lambda \text{ and } \mu\] ,

we must have ,

\[2\lambda + 1 = 5\mu - 1, 3\lambda - 1 = \mu + 2, \lambda = 2\]

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

\[ 3\lambda - \mu = 3 . . . (2)\]

\[ \lambda = 2 . . . (3)\]

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

\[\lambda = 2 \]

\[\mu = 3\]

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

\[LHS = 2\lambda - 5\mu\]

\[ = 2\left( 2 \right) - 5\left( 3 \right)\]

\[ = 4 - 15\]

\[ = - 11 \neq - 2\]

\[ \Rightarrow LHS \neq RHS\]

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