Hello!

We have .

Dividing throughout by , and solving the quadratic in , we get:

, which gives the equations of the pair of straight lines.

The joint equation of bisectors is given by:

(here x in the standard equation found in books is replaced by since the equation of our pair of straight lines is in )

On expanding, we get:

--- (1)

One of the lines represented by the above equation is .

We know that the bisectors of the angles between the lines are perpendicular to each other (Make sketches to convince yourself. Moreover in the joint equation, Coefficient of + Coefficient of = 0 which tells you this fact immediately).

Thus, equation of the other bisector is , where k is the constant to be determined.

(We get the above equation by applying condition of perpendicularity: , where and are the respective slopes. In our case which gives , thus giving the equation of the other bisector)

Multiplying these two equations together,

Comparing with (1),

1) h = 2

2)

So, the other bisector is: