Find the combined equation of the following pairs of lines:Passing thr

1.	Find the combined equation of the following pairs of lines:Passing through (2, 3) and perpendicular to lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0
Answer» Let L₁ and L₂ be the lines passing through the point (2, 3) and perpendicular to the lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0 respectively. Slopes of the lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0 are -3/2 and -1/-3 = 1/3 respectively. ∴ slopes of the lines L₁ and L₂ are 2/3 and -3 respectively. Since the lines L₁ and L₂ pass through the point (2, 3), their equations are y – 3 = \(\frac{2}{3}\)(x – 2) and y – 3 = -3 (x – 2) ∴ 3y – 9 = 2x – 4 and y – 3= -3x + 6 ∴ 2x – 3y + 5 = 0 and 3x – y – 9 = 0 ∴ their combined equation is (2x – 3y + 5)(3x + y – 9) = 0 ∴ 6x² + 2xy – 18x – 9xy – 3y² + 27y + 15x + 5y – 45 = 0 ∴ 6x² – 7xy – 3y² – 3x + 32y – 45 = 0.

Find the combined equation of the following pairs of lines:Passing through (2, 3) and perpendicular to lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0

Answer»

Let L₁ and L₂ be the lines passing through the point (2, 3) and perpendicular to the lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0 respectively.

Slopes of the lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0 are -3/2 and -1/-3 = 1/3 respectively.

∴ slopes of the lines L₁ and L₂ are 2/3 and -3 respectively.

Since the lines L₁ and L₂ pass through the point (2, 3), their equations are

y – 3 = \(\frac{2}{3}\)(x – 2) and y – 3 = -3 (x – 2)

∴ 3y – 9 = 2x – 4 and y – 3= -3x + 6

∴ 2x – 3y + 5 = 0 and 3x – y – 9 = 0

∴ their combined equation is

(2x – 3y + 5)(3x + y – 9) = 0

∴ 6x² + 2xy – 18x – 9xy – 3y² + 27y + 15x + 5y – 45 = 0

∴ 6x² – 7xy – 3y² – 3x + 32y – 45 = 0.