Pair of bisectors of angles
The equation to the pair of bisectors of angles between the pair of straight lines ax
2
+ 2hxy + by
2
= 0 is h(x
2
– y
2
) = (a – b) xy
Proof:
Let ax
2
+ 2hxy + by
2
= 0 represent the pair of lines
y – m
1
x = 0 ----- (1)
y – m
2
x = 0 ----- (2)
The combined equation of (1) & (2) is (y – m
1
x) (y – m
2
x) = 0
⇒ y
2
– (m
1
+ m
2
)xy + m
1
m
2
x
2
= 0
Comparing with
, we have
m
1
+ m
2
=
; m
1
m
2
=
Let α, β be the angles made by the lines (1) & (2) with X-axis.
Let 'θ' be the angle (with X-axis) of the angular bisector of the lines (1) & (2).
Let m
1
, m
2
be the slopes of (1) & (2) respectively and m be the slope of angle bisector.
∴ m
1
= tan α; m
2
= tan β; m = tan θ
From above figure, β – θ = θ – α
⇒ 2 θ = α + β
⇒ tan 2θ = tan(α + β)
⇒ h (x
2
– y
2
) = (a – b)xy