Question

If the real part of \( \frac{z + 1}{z - 1} = 4 \), then the locus of the point representing \( z \) in the complex plane is

• A. a straight line parallel to x-axis
• B. a straight line equally inclined to axes
• C. a circle with radius 2
• D. a circle with radius \( \frac{1}{2} \)

Hint:

Möbius transformations can map lines and circles in the complex plane. In this case, it maps to a circle.

Answer 1

The Correct Option is D

Step 1: Analyze the equation.
The equation \( \frac{z + 1}{z - 1} = 4 \) represents a geometric transformation. This is a Möbius transformation, which typically maps to a circle.
Step 2: Conclusion.
The locus of the point representing \( z \) is a circle with radius \( \frac{1}{2} \). Final Answer: \[ \boxed{\text{a circle with radius } \frac{1}{2}} \]