Bisectors of angle between two given lines:

Say      a₁x+b₁y+c₁ = 0........(1)

a₂x + b₂y + c₂ = 0....(2)  are the 2 intersecting lines.

Let any point p(x, y) be any point on the 2 bisectors of angles of (1) and (2).

Then p is equidistance from (1) and (2)  

that are the required equations of the 2 bisectors of angles between (1) and (2).

If the 2 given lines are not perpendicular that is a₁ a₂ + b₁ b₂ ≠ 0, then one of these equation is the equation of the bisector of acute angle and the other that of the obtuse angle. 

The equation of acute and obtuse angle bisectors:

Method 1

Step 1: Take 1 of the given lines and let the slope of it be m₁ and take one of the bisectors and let it is slope be m₂.

Step 2:  If θ is the acute angle between them, then find the value of 

Step 3:  If tanθ > 1  then bisector taken is the bisector of the obtuse angle and other one will be the bisector of acute angle.

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

Method 2:

     If the constant term c₁ and c₂ in the 2 equations a₁x+b₁y+c₁ = 0 and a₂x + b₂y + c₂ = 0 are having same sign, then

Case 1: if  will be the equation of obtuse angle bisector and

will be the equation of acute angle bisector.

Case 2: if  will give equation of the acute angle bisector and

will give the equation of the obtuse angle bisector.

Note: Whether both lines are perpendicular to each other or not but the angle bisectors of these lines will always be mutually perpendicular.

The equation of bisector of angle which has a given point:

The equation of bisector of angle between the 2 lines containing point (α, β) is 

 are having same signs

or are having opposite signs

The equation of bisector of angle containing the origin:

 Write down the equations of the 2 lines so that constants c₁ and c₂ are positive. Then the equation 

 is the equation of the bisector containing origin.

Note:  if a₁a₂ + b₁b₂ < 0, then origin will lie in acute angle and if a₁a₂ + b₁b₂ > 0 then origin will lie in obtuse angle.

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