Question

The complex number \(z\) which satisfy the equation \(\left|\frac{1+z}{1-z}\right| = 1\) lies on

• A. a circle \(x^2+y^2=1\)
• B. the x-axis
• C. the y-axis
• D. the line \(x+y=1\)

Hint:

\(|z-a| = |z-b|\) $\Rightarrow$ perpendicular bisector of points \(a\) and \(b\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The equation \( |\frac{z - a}{z - b}| = 1 \) represents the locus of a point \( z \) which is equidistant from two points \( a \) and \( b \). This locus is the perpendicular bisector of the line segment joining \( a \) and \( b \).
Step 2: Detailed Explanation:
Rewrite the given equation:
\[ \left| \frac{z - (-i)}{z - i} \right| = 1 \]
\[ |z - (-i)| = |z - i| \]
This means the distance from \( z \) to the point \( (0, -1) \) is equal to the distance from \( z \) to the point \( (0, 1) \).
The points \( (0, 1) \) and \( (0, -1) \) lie on the imaginary axis (y-axis).
The perpendicular bisector of the segment joining \( (0, 1) \) and \( (0, -1) \) is the real axis.
In the Cartesian plane, the real axis is represented by the x-axis (\( y = 0 \)).
Step 3: Final Answer:
The complex number \( z \) lies on the x-axis.