Que: If $arg(\frac{z-z_1}{z-z_2})=\pi$ then what is the locus of $z?$ Doubt In my textbook it is written that it represents the straight line joining $A(Z_1)$ and $B(Z_2)$ but excluding the segment $AB$

IS it right or wrong?

But According to me the locus of $z$ is the straight line joining the line segment $A(Z_1) \ and\ B(Z_2)$


You're right. $$\arg \frac xy = \arg x - \arg y + 2\pi n$$ so $$\arg\frac{z-z_1}{z-z_2}=\pi$$ means the vector $z-z_1$ is parallel to, but points in opposite direction than $z-z_2$.
This implies $z$ is between $z_1$ and $z_2$, hence the answer: 'an interior of the line segment $\overline {z_1 z_2}$'.


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