hello everyone
This is a question on coordinate geometry.
One bisector of the angle between the lines given by a(x-1)^2 + 2h(x-1)y + by^2 = 0 is 2x + y – 2 = 0. We need to find the other bisector.
Any help would be appreciated. Thanks in advance.
hello everyone
This is a question on coordinate geometry.
One bisector of the angle between the lines given by a(x-1)^2 + 2h(x-1)y + by^2 = 0 is 2x + y – 2 = 0. We need to find the other bisector.
Any help would be appreciated. Thanks in advance.
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:
Alternatively,
we can find the point of intersection of the angle bisector 2x+y-2 with the lines
and find the equation of the perpendicular to the bisector that goes through
that same point of intersection.
Hence, y=2-2x is substituted
then
x=1
The slope of is -2, as
Any perpendicular line has slope
The perpendicular bisector contains the point (1,0)
hence it's equation is