Question

Find the equation of the bisectors of the angle between the lines:
$y-b=\frac{2m}{1-m^2}(x-a)$ and $y-b=\frac{2M}{1-M^2}(x-a)$

Answer 1

If $m=\tan \theta ,M=\tan \varphi$
$\frac{2m}{1-m^2}=\tan 2\theta$
Both the lines pass through $(a,b)$ and are inclined to axis of $x$ at an angle $2\theta ,2\varphi$ respectively
Hene the bisector of these lines will make an angle $\theta +\varphi$ with x-axis so that its slope will be
$\tan (\theta +\varphi )=\frac{m+M}{1-mM}=M_1$
The other bisector will be perpendicular to it whose slope will be $\frac{1-mM}{m+M}=M_2$
Both these pass throguh $(a,b)$. hence their equations are $(y-b)=M_1(x-a)$ and $(y-b)=M_2(x-a)$