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Question
Determine whether the following pair of lines intersect or not:
\[\frac{x - 5}{4} = \frac{y - 7}{4} = \frac{z + 3}{- 5} and \frac{x - 8}{7} = \frac{y - 4}{1} = \frac{3 - 5}{3}\]
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Solution
\[\frac{x - 5}{4} = \frac{y - 7}{4} = \frac{z + 3}{- 5} and \frac{x - 8}{7} = \frac{y - 4}{1} = \frac{3 - 5}{3}\]
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
\[4\lambda + 5 = 7\mu + 8, 4\lambda + 7 = \mu + 4, - 5\lambda - 3 = 3\mu + 5\]
\[ \Rightarrow 4\lambda - 7\mu = 3 . . . (1)\]
\[ 4\lambda - \mu = - 3 . . . (2)\]
\[ 5\lambda + 3\mu = - 8 . . . (3)\]
\[\text{ From (1) and (2), we get }\]
\[\lambda = - 1 \]
\[\mu = - 1\]
\[\text{ Substituting } \lambda = - 1 \text{ and \mu = - 1 in (3), we get} \]
\[LHS = 5\lambda + 3\mu \]
\[ = 5\left( - 1 \right) + 3\left( - 1 \right)\]
\[ = - 8\]
\[ = RHS\]
\[\text{ Since } \lambda = - 1 \text{ and \mu = - 1 satisfy (3), the lines intersect } . \]
\[\text{ Substituting } \lambda = - 1 \text{ and \mu = - 1 in the coordinates of a general point on the first line, we get } \]
\[x = 1\]
\[y = 3\]
\[z = 2\]
\[\text{ Hence, the given lines intersect at } \left( 1, 3, 2 \right) .\]
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