Question

Find the combined equation of the following pairs of lines:

Passing through (-1, 2),one is parallel to x + 3y – 1 = 0 and the other is perpendicular to 2x – 3y – 1 = 0.

Answer 1

Let L₁ be the line passing through (-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 L₂ be the line passing through (-1, 2) and perpendicular to the line 2x – 3y – 1 = 0

whose slope is \(\frac{-2}{-3}\) = \(\frac{2}{3}\)

∴ slope of the line L₂ is – \(\frac{3}{2}\)

∴ equation of the line L₂ is

y – 2= – \(\frac{3}{2}\) (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