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Question
Find the combined equation of the following pair of line passing through (−1, 2), one is parallel to x + 3y − 1 = 0 and other is perpendicular to 2x − 3y − 1 = 0
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Solution
Let L1 be the line passing through the point (−1, 2) and parallel to the line x + 3y − 1 = 0 whose slope is `-1/3`.
∴ slope of the line L1 is `-1/3`
∴ equation of the line L1 is
y − 2 = `- 1/3` (x + 1)
∴ 3y − 6 = − x − 1
∴ x + 3y − 5 = 0
Let L1 be the line passing through (−1, 2) and perpendicular to the line 2x − 3y − 1 = 0 whose slope is `(-2)/-3 = 2/3`
∴ slope of the line L2 is `- 3/2`
∴ equation of the line L2 is
y - 2 = `-3/2`(x + 1)
∴ 2y − 4 = − 3x − 3
∴ 3x + 2y − 1 = 0
Hence, the equations of the required lines are
x + 3y − 5 = 0 and 3x + 2y − 1 = 0
∴ their combined equation is
(x + 3y − 5)(3x + 2y − 1) = 0
∴ 3x2 + 2xy − x + 9xy + 6y2 − 3y − 15x − 10y + 5 = 0
∴ 3x2 + 11xy + 6y2 − 16x −13y + 5 = 0
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