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Question

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
d1-c1d2-c2a1b2-a2b1 sq. units.
Deduce the condition for these lines to form a rhombus.

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Solution

The given lines are

a1x + b1y + c1 = 0 ... (1)

a1x + b1y + d1 = 0 ... (2)

a2x + b2y + c2 = 0 ... (3)

a2x + b2y + d2 = 0 ... (4)

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=c1-d1c2-d2a1a2b1b2

a1a2b1b2=a1b2-a2b1

Area=c1-d1c2-d2a1b2-a2b1=d1-c1d2-c2a1b2-a2b1

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

c1-d1a12+b12=c2-d2a22+b22

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