Question

The lines x = ay + b, z = cy + d and x = a'y + b', z = c'y + d' are perpendicular to each other, if ______

Options

• aa' + cc' = 1

• aa' + cc' = -1

• ac + a'c' = 1

• ac + a'c' = -1

Answer 1

The lines x = ay + b, z = cy + d and x = a'y + b', z = c'y + d' are perpendicular to each other, if aa' + cc' = -1.

Explanation:

Given equations of lines are

x = ay + b, z = cy + d

⇒ `(x - b)/a = y/1, (z - d)/c = y/1`

⇒ `(x - b)/a = y/1 = (z - d)/c`

and x = a'y + b', z = c'y + d'

⇒ `(x - b^')/a^' = y/1 = (z - d^')/c^'= y/1`

⇒ `(x - b^')/a^' = y/1 = (x - d^')/c^'`

Since the lines are perpendicular to each other,

a₁a₂ + b₁b₂ + c₁c₂ = 0

⇒ >aa' + 1(1) + cc' = 0

⇒ aa' + cc' = -1