complex geometry

Question

Give any two complex numbers lie along the angle bisector of the line L1 : z= (1+ 3r) + i(1+4r) L2 : z =(1+3s) + i(1-4s) find any two points on the angle bisector

Answer 1

An interesting problem.    The two lines intersect at the point   1 + i  (where r and s are both zero)

 

It would seem useful to visualize the evolution of the lines keeping r = s. 

The real parts are both  1 + 3r , whereas the imaginary parts diverge with equal and opposite slopes ( 4 and -4)

 

I think this means the the bisector is the real valued line   z = 1 + 3 r

 

Thus two points on the bisector could be  the real numbers 1 and 4