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

Answer 1

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