Prove that the area of the parallelogram formed by the lines a 1 x + b 1 y + c 1 = 0, a 1 x + b 1 y + d 1 = 0, a 2 x + b 2 y + c 2 = 0, a 2 x + b 2 y + d 2 = 0 is sq. units. Deduce the condition for these lines to form a rhombus.

Question

Philoid · Accepted Answer

Given:

The given lines are

a₁x + b₁y + c₁ = 0 … (1)

a₁x + b₁y + d₁ = 0 … (2)

a₂x + b₂y + c₂ = 0 … (3)

a₂x + b₂y + d₂ = 0 … (4)

To prove:

The area of the parallelogram formed by the lines

a₁x + b₁y + c₁ = 0, a₁x + b₁y + d₁ = 0, a₂x + b₂y + c₂ = 0, a₂x + b₂y + d₂ = 0 is sq. units.

Explanation:

The area of the parallelogram formed by the lines a₁x + b₁y + c₁ = 0, a₁x + b₁y + d₁ = 0, a₂x + b₂y + c₂ = 0 and a₂x + b₂y + d₂ = 0 is given below:

Area

∵a₁b₂ – a₂b₁

∴ Area

If the given parallelogram is a rhombus, then the distance between the pair of parallel lines are equal.

∴

Hence proved.

Prove that the area of the parallelogram formed by the linesa1x + b1y + c1 = 0, a1x + b1y + d1 = 0, a2x + b2y + c2 = 0, a2x + b2y + d2 = 0 is sq. units.Deduce the condition for these lines to form a rhombus.

Prove that the area of the parallelogram formed by the lines
a₁x + b₁y + c₁ = 0, a₁x + b₁y + d₁ = 0, a₂x + b₂y + c₂ = 0, a₂x + b₂y + d₂ = 0 is sq. units.

Deduce the condition for these lines to form a rhombus.