Question

show that the complex no.s satisfying the condition arg (z-1 / z+1)=pi/4 lies on a circle

Answer 1

Hi,
arg((z−1)/(z+1))=π/4 ⟹ arg(z−1)−arg(z+1)=π/4 … (i)

z−1 is represented by line from 1 to z and z+1 by line from −1 to z

For (i) to be true arg(z−1) > arg(z+1) so z must be in upper half of plane. Also by angle in same
segment theorem z must lie on the circle thro 1 and −1 that contains π/4 in its upper segment.

For the lower part of this circle arg((z−1)/(z+1))=−3π/4 (5π/4)

Since the argument of the complex number zero is undefined z=±1 are excluded from the locus.
So |z|=1
x²+y²= 1
It is a circle.
Regards