Question

The locus of the complex number \( Z \) satisfying \( \arg \left( \frac{Z - 1}{Z + 1} \right) = \frac{\pi}{4} \) is:

• A. A straight line
• B. A circle
• C. A parabola
• D. An ellipse

Hint:

For complex number loci, use transformations to convert into Cartesian equations.

Answer 1

The Correct Option is B

Step 1: Convert to Cartesian Form Let \( Z = x + iy \), then \[ \frac{Z - 1}{Z + 1} = \frac{x + iy - 1}{x + iy + 1} \] Taking argument on both sides: \[ \arg \left( \frac{Z - 1}{Z + 1} \right) = \frac{\pi}{4} \]
Step 2: Convert into Locus Equation Expanding using properties of arguments: \[ \tan^{-1} \left( \frac{y}{x-1} \right) - \tan^{-1} \left( \frac{y}{x+1} \right) = \frac{\pi}{4} \] Solving further, we obtain: \[ (x-1)^2 + y^2 = 1 \] which represents a circle centered at (1,0) with radius 1. Thus, the correct answer is: \[ \boxed{\text{A Circle}} \]