# Show that the diagonals of the parallelogram whose sides are lx + my + n = 0, lx + my + n’ = 0, mx + ly + n = 0 and mx + ly + n’ = 0 include an angle.

Given:

The given lines are

lx + my + n = 0 … (1)

mx + ly + n’ = 0 … (2)

lx + my + n’ = 0 … (3)

mx + ly + n = 0 … (4)

To prove:

The diagonals of the parallelogram whose sides are lx + my + n = 0, lx + my + n’ = 0, mx + ly + n = 0 and mx + ly + n’ = 0 include an angle.

Explanation:

Solving (1) and (2), we get,

B

Solving (2) and (3), we get,

C

Solving (3) and (4), we get,

D

Solving (1) and (4), we get,

A

Let m1 and m2 be the slope of AC and BD.

m1m2 = -1

Hence proved, diagonals of the parallelogram intersect at an angle

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