If arg ((z-1)/(z+1))=(π)/(2), where z is a complex number, locus of z is (a) |z|=1, Im(z)>0 (b)

step 1 In this question, it is given that argument z minus 1 upon z plus 1 is equal to pi by 2. We need

If arg ((z-1)/(z+1))=(π)/(2), where z is a complex number,...

If arg $\left(\frac{z-1}{z+1}\right)=\frac{\pi}{2}$, where $z$ is a complex number, locus of $z$ is
(a) $|\mathrm{z}|=1, \operatorname{Im}(\mathrm{z})>0$
(b) $|\mathrm{z}|=1, \operatorname{Im}(\mathrm{z})<0$
(c) $|z|=1$
(d) $|z-1|=1, \operatorname{Im}(z)>0$

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Key Concepts

Complex Numbers

Complex numbers are expressions of the form a + bi, where a represents the real part and b represents the imaginary part. They form the basis of analysis and geometry in the complex plane, allowing problems to be interpreted both algebraically and geometrically.

Argument of a Complex Number

The argument of a complex number is the angle made with the positive real axis in the complex plane. It is essential for understanding the direction of a complex number and plays a crucial role when comparing or computing ratios of complex numbers.

Locus in the Complex Plane

Determining a locus in the complex plane involves finding the set of all points (complex numbers) that satisfy a given condition or equation. This concept is fundamental when analyzing geometric representations of complex equations, such as circles or lines.

Möbius Transformation

A Möbius transformation, generally expressed as (az+b)/(cz+d), is a mapping of the complex plane that preserves angles and transforms circles and lines into circles or lines. Recognizing and manipulating such transformations aids in uncovering the geometric nature embedded within complex equations.

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