Find the equation of angle bisector of 2lines passing through quadrant containing$(2,3)$

Question

Question - $L_1⇒2x+y-1=0$ , $L_2⇒2x-y+3=0$. Find the equation of angle bisector passing through quadrant containing $(2,3).$

Effort $⇒$ Found out first angle bisector $b_1⇒y=2$ , second angle bisector $b_2⇒-\dfrac{1}{2}$

Parity check for $L_1 ⇒2(2)+3-1=6>0$

Parity check for $L_2⇒2(2)-(3)+3=4>0$

Therefore $(2,3)$ is in $(+,+)$ with respect to $L_1$and $L_2$.

What to do next? I am stuck here , can someone help?I only know that we check $(0,0)$ after this. But how to get angle bisector after that? Thanks.

Answer 1

Start with finding the point of intersection: $y=1-2x, y=2x+3$. Point of intersection is $(-\frac{1}{2},2)$. Note that the slopes are opposite so one bisector is vertical and one is horizontal. The required bisector is $y=2$.