Question

Find the combined equation of pair of lines passing through (2, 3) and perpendicular to the lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0.

Answer 1

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.

∴ Sopes 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 = 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.