Question

Determine whether the following pair of lines intersect or not:  

\[\frac{x - 5}{4} = \frac{y - 7}{4} = \frac{z + 3}{- 5} and \frac{x - 8}{7} = \frac{y - 4}{1} = \frac{3 - 5}{3}\]

Answer 1

\[\frac{x - 5}{4} = \frac{y - 7}{4} = \frac{z + 3}{- 5} and \frac{x - 8}{7} = \frac{y - 4}{1} = \frac{3 - 5}{3}\] 

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

\[\frac{x - 5}{4} = \frac{y - 7}{4} = \frac{z + 3}{- 5} = \lambda\]

\[ \Rightarrow x = 4\lambda + 5\]

\[ y = 4\lambda + 7 \]

\[ z = - 5\lambda - 3\]

The coordinates of a general point on the first line are

\[\left( 4\lambda + 5, 4\lambda + 7, - 5\lambda - 3 \right)\]

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

\[\frac{x - 8}{7} = \frac{y - 4}{1} = \frac{z - 5}{3} = \mu\]

\[ \Rightarrow x = 7\mu + 8\]

\[ y = \mu + 4 \]

\[ z = 3\mu + 5\]

The coordinates of a general point on the second line are

\[\left( 7\mu + 8, \mu + 4, 3\mu + 5 \right)\]

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

\[\lambda \text{ and }  \mu\] we must have

\[4\lambda + 5 = 7\mu + 8, 4\lambda + 7 = \mu + 4, - 5\lambda - 3 = 3\mu + 5\]

\[ \Rightarrow 4\lambda - 7\mu = 3 . . . (1)\]

\[ 4\lambda - \mu = - 3 . . . (2)\]

\[ 5\lambda + 3\mu = - 8 . . . (3)\]

\[\text{ From (1) and (2), we get }\]

\[\lambda = - 1 \]

\[\mu = - 1\]

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

\[LHS = 5\lambda + 3\mu \]

\[ = 5\left( - 1 \right) + 3\left( - 1 \right)\]

\[ = - 8\]

\[ = RHS\]

\[\text{ Since } \lambda = - 1 \text{ and \mu = - 1 satisfy (3), the lines intersect } . \]

\[\text{ Substituting } \lambda = - 1 \text{ and \mu = - 1 in the coordinates of a general point on the first line, we get } \]

\[x = 1\]

\[y = 3\]

\[z = 2\]

\[\text{ Hence, the given lines intersect at }  \left( 1, 3, 2 \right) .\]