Question

Find the values of p and q if the equation px² – 6xy + y² + 18x – qy + 8 = 0 represents a pair of parallel lines.

Answer 1

Given equation:

px² – 6xy + y² + 18x – qy + 8 = 0

Comparing the given equation with

ax² + 2hxy + by² + 2gx + 2fy + c = 0, we get

a = p, b = 1, c = 8, h = –3, g = 9, f = `(-q)/2`

For parallel lines h² – ab = 0

∴ (–3)² – (p)(1) = 0

∴ 9 – p = 0

∴ p = 9

The general second-degree equation represents a pair of lines if

abc + 2fgh – af² – bg² – ch² = 0

∴ `(p)(1)(8) + 2(-q/2)(9)(-3) - (p)(-q/2)^2 - (1)(9)^2 - (8)(-3)^2` = 0

Put p = 9

∴ `72 + 27q - 9/4q^2 - 81 - 72` = 0

∴ `9/4q^2 - 27q + 81` = 0

Dividing by 9

`q^2/4 - 3q + 9` = 0

∴ q² – 12q + 36 = 0

∴ (q – 6)² = 0

∴ q – 6 = 0

∴ q = 6

∴ p = 9, q = 6