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 a_{1}x + b_{1}y + c_{1} =
0 and a_{2}x + b_{2}y + c_{2} = 0 are

If the two given lines are not
perpendicular i.e. a_{1} a_{2} + b_{1 }b_{2} >
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.

Take one of the lines
and let its slope be m 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 a_{1} x + b_{1}y + c_{1} = 0 and a_{2}x+
b_{2}y + c_{2} = 0, then

=

will
represent the equation of the bisector of the acute or obtuse angle between the
lines according as c_{1}c_{2}(a_{1}a_{2} + b_{1}b_{2})
is negative or positive. .

Write the equations of
the two lines so that the constants c_{1} and c_{2} become
positive. Then the equation is
the equation of the bisector containing the origin.

Remark:

(i)
If a_{1}a_{2}
+ b_{1}b_{2} < 0, then the origin will lie in the acute
angle and if

a_{1}a_{2} + b_{1}b_{2} > 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 between the two lines containing the point (a,b)
is or
according
as

a_{1}a + b_{1}b
+ c_{1} and a_{2} a + b_{2}b
+ c_{2}^{ }are of the same signs or of opposite signs.

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).

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

Making C_{1} and C_{2}
positive in the given equations, we get

–4x – 3y + 6 = 0 and 5x + 12y + 9 = 0

Since a_{1}a_{2} + b_{1}b_{2}
= –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.

Let
L Let
L L Let
(h, k) be a point on L = . |

Since (h, k) lies on L_{2}, a_{2}h + b_{2}k
+ c_{2} = 0

&⇒ a_{1}^{2}
+ a_{2}^{2} l^{2} + 2a_{1}a_{2}l
+ b_{1}^{2} + b_{2}^{2}l^{2} + 2b_{1}b_{2}l
= a_{1}^{2} + b_{1}^{2}

&⇒ l
= 0 or

But l = 0 gives L_{3}
= L_{1} . Hence L_{3} º .

Note:

Some
times the reflected ray L_{3} is also called the mirror image of L_{1}
in L_{2} .