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Question
Determine whether the following pair of lines intersect or not:
\[\frac{x - 1}{2} = \frac{y + 1}{3} = z \text{ and } \frac{x + 1}{5} = \frac{y - 2}{1}; z = 2\]
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Solution
\[\frac{x - 1}{2} = \frac{y + 1}{3} = z \text{ and } \frac{x + 1}{5} = \frac{y - 2}{1}; z = 2\]
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 ,
\[2\lambda + 1 = 5\mu - 1, 3\lambda - 1 = \mu + 2, \lambda = 2\]
\[ \Rightarrow 2\lambda - 5\mu = - 2 . . . (1)\]
\[ 3\lambda - \mu = 3 . . . (2)\]
\[ \lambda = 2 . . . (3)\]
\[\text{ Solving (2) and (3), we get } \]
\[\lambda = 2 \]
\[\mu = 3\]
\[\text{ Substituting } \lambda = 2 \text{ and } \mu = 3 \text{ in} \left( 1 \right), \text{ we get } \]
\[LHS = 2\lambda - 5\mu\]
\[ = 2\left( 2 \right) - 5\left( 3 \right)\]
\[ = 4 - 15\]
\[ = - 11 \neq - 2\]
\[ \Rightarrow LHS \neq RHS\]
\[\text{ Since} \lambda = 2 \text{ and } \mu = 3 \text{ do not satisfy (1), the given lines do not intersect } .\]
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