Question

One of the bisector of the angle between the lines a(x-1)^2 + 2h(x-1)(y-2) + b(y-2)^2 =0. is x+2y=5. The other bisector is ?

Answer 1

Given that one of the bisector of the angle between the lines
`a(x-1)^2 + 2h(x-1)(y-2) + b(y-2)^2=0`. is `x+2y=5`.

The given equation of the pair of sraight line `a(x-1)^2 + 2h(x-1)(y-2) + b(y-2)^2=0` suggests that the pair of straight lines must intersect at `(1,2)` and this also satisfies the equation of the given bisector `x+2y=5`.

The other bisector will be normal to the given bisector and will pass through `(1,2)`. The equation of any straight line perpendicular to the given bisector is `2x-y=c`.
This also passes through `(1,2)`

So `2*1-2=c=>c=0`

Hence the equation of the other bisector will be

`2x-y=0`

Answer 2

The other bisector is `2x-y=0`.

Explanation:

Let us apply translation of coordinate axes using `x-1=x'` and `y-2=y'` (this is equivalent of moving origin to `(1,2)`), then the equation becomes

`ax'^2+2hx'y'+by'^2=0`

which represents a pair of lines `y'=mx'` and `y=nx'`, where `m!=n`.

Observe that the lines intersected at `(1,2)` and now they intersect at `(0,0)`. Further, bisectors of angles are at right angles to each other.

Let us transform the equation of given bisector `x+2y=5` in new axes. As `x=x'+1` and `y=y'+2`, the revised equation of bisector is `x'+2y'=0` and as in slope intercept form, it is `y'=-1/2x'`, its slope is `-1/2`.

Now as bisectors of angles are are ar right angles to each other, slope of other bisector is `-1/(-1/2)=2` and its equation will be

`y'=2x'`

Substituting `y'` and `x'`,

`y-2=2(x-1)` or `2x-y=0`

graph{(2x-y)(x+2y-5)=0 [-9.88, 10.12, -3, 7]}

Observe that we cannot have equations of original pair of intersecting lines as tere could be infinite possibilities.