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Question
Classify the following pair of line as coincident, parallel or intersecting:
3x + 2y − 4 = 0 and 6x + 4y − 8 = 0.
Solution
Let \[a_1 x + b_1 y + c_1 = 0 \text { and } a_2 x + b_2 y + c_2 = 0\]
(a) The lines intersect if \[\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\] is true.
(b) The lines are parallel if \[\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\] is true.
(c) The lines are coincident if \[\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\] is true.
3x + 2y − 4 = 0 and 6x + 4y − 8 = 0
Here,
\[\frac{3}{6} = \frac{2}{4} = \frac{- 4}{- 8}\]
Therefore, the lines 3x + 2y − 4 = 0 and 6x + 4y − 8 = 0 are coincident.
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