If |(z - i)/(z + i) | = 1, then find the locus of z — JEE Mathematics

The given equation can be written as:

$$|z - i| = |z + i|$$

This equation represents the set of all points $z$ that are equidistant from the points $i$ (0, 1) and $-i$ (0, -1).
The locus of points equidistant from two given points is the perpendicular bisector of the line segment joining them.
The line segment joins $(0,1)$ and $(0,-1)$, and its perpendicular bisector is the x-axis ($y = 0$).

Alternatively, let $z = x + iy$:

$$|x + i(y - 1)| = |x + i(y + 1)|$$

$$x^2 + (y - 1)^2 = x^2 + (y + 1)^2$$

$$-2y = 2y \implies 4y = 0 \implies y = 0$$

Thus, the locus of $z$ is the Real axis (x-axis).

Answer: Real axis ($Im(z) = 0$)