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