Question

The equation of the bisector of the angle between the lines $2x+y-6=0$ and $2x-4y+7=0$ containing the point $(1,2)$ is

• A. $6x+2y+5=0$
• B. $6x-2y+5=0$
• C. $6x-2y-5=0$
• D. $2x+6y-19=0$

Answer 1

The correct option is C $6x-2y-5=0$

The equation of the bisector of the angle between the lines $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ are

$\frac{a_{1}x+b_{1}y+c_1}{\sqrt{a_1^2+b_1^2}}=\pm \frac{a_{2}x+b_{2}y+c_2}{\sqrt{a_2^2+b_2^2}}$

Then, the equation of the bisectors between the given lines are
$\frac{2x+y-6}{\sqrt{5}}=\pm \frac{2x-4y+7}{2\sqrt{5}}$
i.e., $2x+6y-19=0$ and $6x-2y-5=0$.
Substituting $(1,2)$ on the LHS of the equations of the two given lines, we see that
$2+12-19=-5<0$ and $6-12-5<0$
Therefore, the equation which is obtained by taking negative sign is the equation of bisector containing the point $(1,2)$ is
$6x-2y-5=0$