Bisectors of the Angles between two given lines

Angular bisector is the locus of a point which moves in such a way so that its distance from two intersecting lines remains same. .

The equations of the two bisectors of the angles between the lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are

If the two given lines are not perpendicular i.e. a₁ a₂ + b₁b₂ > 0, then one of these equations is the equation of the bisector of the acute angle and the other that of the obtuse angle.

Note:

Whether both lines are perpendicular or not but the angular bisectors of these lines will always be mutually perpendicular.

The bisectors of the acute and the obtuse angles:

Take one of the lines and let its slope be m₁ and take one of the bisectors and let its slope be m₂. If q be the acute angle between them, then find

If tanq > 1 then the bisector taken is the bisector of the obtuse angle and the other one will be the bisector of the acute angle.

If 0 < tanq < 1 then the bisector taken is the bisector of the acute angle and the other one will be the bisector of the obtuse angle.

If two lines are a₁ x + b₁y + c₁ = 0 and a₂x+ b₂y + c₂ = 0, then

will represent the equation of the bisector of the acute or obtuse angle between the lines according as c₁c₂(a₁a₂ + b₁b₂) is negative or positive. .

The equation of the bisector of the angle containing the origin:

Write the equations of the two lines so that the constants c₁ and c₂ become positive. Then the equation is the equation of the bisector containing the origin.

Remark:

(i) If a₁a₂ + b₁b₂ < 0, then the origin will lie in the acute angle and if
a₁a₂ + b₁b₂ > 0, then origin will lie in the obtuse angle.

(ii) The remark (i) is helpful in finding the equation of bisector of the obtuse angle or acute angle directly.

The equation of the bisector of the angle which contains a given point:

The equation of the bisector of the angle between the two lines containing the point (a,b) is or according as
a₁a + b₁b + c₁ and a₂ a + b₂b + c₂are of the same signs or of opposite signs.

Example 1

For the straight lines 4x + 3y – 6 = 0 and 5x + 12y + 9 = 0, find the equation of the

bisector of the obtuse angle between them.
bisector of the acute angle between them.
bisector of the angle which contains (1, 2).

Solution:

Equations of bisectors of the angles between the given lines are

&⇒ 9x – 7y – 41 = 0 and 7x + 9y – 3 = 0

If q is the acute angle between the line 4x + 3y – 6 = 0 and the bisector 9x – 7y – 41 = 0, then tan q =

Hence

The bisector of the obtuse angle is 9x – 7y – 41 = 0.
The bisector of the acute angle is 7x + 9y – 3 = 0 The bisector of the angle containing the origin
&⇒ 7x + 9y – 3 = 0.
For the point (1, 2), 4x + 3y – 6 = 4 ´ 1 + 3´2 – 6 > 0

5x + 12y + 9 = 12´2 + 9 > 0

Hence equation of the bisector of the angle containing the point
(1, 2) is
&⇒ 9x – 7y – 41 = 0

Alternative:

Making C₁ and C₂ positive in the given equations, we get

–4x – 3y + 6 = 0 and 5x + 12y + 9 = 0

Since a₁a₂ + b₁b₂ = –20 –36 = –56 < 0, so the origin will lie in the acute angle. Hence bisector of the acute angle is given by.

i.e. 9x – 7y – 41 = 0

Similarly bisector of obtuse angle is 7x + 9y – 3 = 0.

The equation of reflected ray :

Let L₁ º a₁x + b₁y + c₁ = 0 be the incident ray in the line mirror L₂ º a₂x + b₂y + c₂ = 0.

Let L₃ be the reflected ray from the line L₂. Clearly L₂ will be one of the bisectors of the angles between L₁ and L₃. Since L₃ passes through A, so .

L₃ º L₁ + lL₂ = 0.

Let (h, k) be a point on L₂. Then, .

= .