Equation of the Bisectors

To find the equation of the bisectors of the angle between ax+by+c=0 and ax₁+by₁+c₁=0.

Let the lines AB and CD intersect at S.

Let P(x,y) be a point on the angle bisector of any one of the angles. Let P(x,y) be on SU, then the length PM of the perpendicular drawn from P on AB = PL, the length drawn to CD.

The origin and the point P are on the same side of both the lines SA and SD.

ax + by + c and a₁x + b₁y + c₁ will both be positive as C and C₁ are both positive.

Now the above ratio is true for all points on SU.

The equation of the angle bisector SU is

............ ( i )

Here O and P are on the opposite sides of SD. Therefore C and

ax + by + c must be of opposite signs. But C > 0, ax + by + c < 0

or -(ax + by + c) > 0. Again, since O and P are on the same side of the line

The point P(x,y) lies on the bisector ST.

The length of PQ, the perpendicular drawn from P on SD = perpendicular PR drawn from P onto SB.

The relation holds good for all points on the bisector ST, therefore the equation of the bisectors ST is

Using (i) and (ii), the equation of the bisectors of the angles can be written as