how to find the equations of the lines if the equation of their angle bisectors are given and one of the line passes through a given point?
Let the two lines be 1.)ax+by+c=0; and 2.)dx+ey+f=0 ;
then the 2 angle bisectors are obtained by the formula;
(ax+by+c)/_/(a^2+b^2) ={+-} (dx+ey+f)/_/(d^2+e^2) ...........(A.)
so we get equations in a,b,c,d,e,f by comparing the equations of the given angle bisectors with the equations obtained from ...(A.). Also, let the line (1.) pass through the given point (h,k); then ah+bk+c=0. So,we get another equation in a,b,c. Solving the above equations; we get values of a,b,c,d,e,f and hence the lines (1.) and (2.).
Please note:- the symbol " _/ " means "square root".