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: \[ \boxed{\text{A Circle}} \]

Answer 2

To determine the locus of the complex number \( Z \) that satisfies the condition \( \arg \left( \frac{Z - 1}{Z + 1} \right) = \frac{\pi}{4} \), we can start by breaking down this expression.

Let \( Z = x + yi \), where \( x \) and \( y \) are real numbers. Then, we can express: 

\( \frac{Z - 1}{Z + 1} = \frac{x + yi - 1}{x + yi + 1} \).

Write \( \frac{Z - 1}{Z + 1} = u + vi \), where \( u \) and \( v \) are real.

The argument condition \( \arg \left( \frac{Z - 1}{Z + 1} \right) = \frac{\pi}{4} \) implies that:

\( \tan^{-1} \left(\frac{v}{u}\right) = \frac{\pi}{4} \), hence \( \frac{v}{u} = 1 \) or \( v = u \).

So, the imaginary part is equal to the real part for the complex number \( \frac{Z - 1}{Z + 1} \).

To simplify further, equate the imaginary and real parts:

\(\text{Real: } \frac{(x-1)(x+1) + y^2}{(x+1)^2+y^2}\)
\(\text{Imaginary: } \frac{y(x+1) - y}{(x+1)^2+y^2}\)

This means:

\( (x-1)(x+1) + y^2 = y(x+1) - y \).

Solving this results in:

\( (x^2 - 1 + y^2) = y \).

This can be rewritten as:

\( x^2 + y^2 - y = 1 \).

To complete the square for \( y \), rewrite it as:

\( x^2 + (y - \frac{1}{2})^2 = \frac{5}{4} \).

This is the equation of a circle with center at \( \left( 0, \frac{1}{2} \right) \) and radius \( \frac{\sqrt{5}}{2} \).

Hence, the locus of \( Z \) is a circle.