$A=\bigl\{z:\arg\left(\frac{z-1}{z}\right)=\frac{π}{2}\bigr\}$ and $B=\{z:\arg\left(z-1\right)=π\}$. Find $(A\;\cap\;B)$.

Question

A & B are sets of complex numbers

Since $\arg(z-1)=π$, therefore $z$ is a purely real complex number. Thus, $\arg\left(\frac{z-1}{z}\right)$ has to be equal to $0$ or $π$.

So, $A \cap B=\varnothing$.

Is my approach correct?

Answer 1

A={z:$arg(\frac{z-1}{z})=\frac{π}{2}$} $$ $$ Locus of z is semicircle above x axis with (0,0) and (1,0) as one of its diameter and z$\ne$0 , 1. $$. $$ B = {z: arg(z-1)=π } $$. $$ Locus of z is a ray from z=1 towards -ve direction of x axis where z$\ne$1. $$. $$ Hence no z satisfy both locus $$ $$ For more on complex number you can refer to https://www.mathsdiscussion.com/best-iitjee-maths-for-mains-and-advance/