Question

One bisector of the angle between the lines given by $a{(x - 1)^2} + 2h(x - 1)y + b{y^2} = 0$ is $2x + y - 2 = 0$ .The other bisector is.
A. $x - 2y + 1 = 0$
B. $2x + y - 1 = 0$
C. $x + 2y - 1 = 0$
D. $x - 2y - 1 = 0$

Answer 1

Hint: One bisector of the angle between the lines given by $a{(x - 1)^2} + 2h(x - 1)y + b{y^2} = 0$ is $2x + y - 2 = 0$ the other bisector will be perpendicular to the given bisector. The equation of the other bisector is given by $x - 2y + \lambda = 0$ . First of all find the points of intersection and then put that point into the general equation of the bisector.

Complete step-by-step answer:
The bisector of the angle between the lines is at right angles it means if the equation of one bisector is $2x + y - 2 = 0$ then equation of other bisector will be of the form $x - 2y + \lambda = 0$
$a{(x - 1)^2} + 2h(x - 1)y + b{y^2} = 0$ Equation suggest that the pair of straight line must intersect at point (1, 0) and it also satisfies the given bisector equation. The points are found as follows: -
Equate the variable terms of the equation to zero.
$ \Rightarrow x - 1 = 0$
$\therefore x = 1$
$\therefore y = 0$
Bisector given is $2x + y - 2 = 0$ which the equation of line is. Point (1, 0) satisfies this bisector equation.
$ \Rightarrow 2(1) + 0 - 2$
$ \Rightarrow 2(1) + 0 - 2 = 0$ (Putting value of x and y in the equation)
This shows that points satisfy the equation.
The other bisector will pass through the same point (1, 0) and will be perpendicular to the given bisector. And it will be of the form $x - 2y + \lambda = 0$
Putting x=1 and y=0 in $x - 2y + \lambda = 0$
$ \Rightarrow 1 - 2(0) + \lambda = 0$
$ \Rightarrow 1 + \lambda = 0$
$\therefore \lambda = - 1$ Putting this lambda value into the bisector equation $x - 2y + \lambda = 0$
$ \Rightarrow x - 2y + ( - 1) = 0$
$\therefore x - 2y - 1 = 0$
Hence the other bisector equation is $x - 2y - 1 = 0$

So, the correct answer is “Option D”.

Note: Students must remember the following equations:-
Equation of line $\Rightarrow $ $y = mx + c$ where x, y are the coordinates on x-axis and y-axis while m is the slope of the line and c is the intercept.
If two points are given for example $({x_1},{x_2})$ and $({y_1},{y_2})$ then slope is given by $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$