Question

Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.

Answer 1

Given that, x = py + q

⇒ y = `(x - "q")/"p"`

And z = ry + s

⇒ y = `(z - "s")/"r"`

∴ The equation becomes `(x - "q")/"p" = y/1 = (z - "s")/"r"` in which d’ratios are a₁ = p, b₁ = 1, c₁ = r

Similarly x = p'y + q' 

⇒ y = `(x - "q'")/"p'"`

And z = r'y + s' 

⇒ y = `(z - "s'")/"r'"`

∴ The equation becomes `(x - "q'")/"p'" = y/1 = (z - "s'")/"r'"` in which a₂ = p', b₂ = 1, c₂ = r'

If the lines are perpendicular to each other, then

a₁a₂ + b₁b₂ + c₁c₂ = 0

pp' + 1.1 + rr' = 0

Hence, pp' + rr' + 1 = 0 is the required condition.