Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + iy$, where $x$ and $y$ are real numbers and $i = \sqrt{-1}$, the modulus is $|z| = \sqrt{x^2 + y^2}$. Geometrically, it represents the distance from the origin to the point $(x, y)$ in the complex plane.
• Modulus of a Quotient: For complex numbers $z_1$ and $z_2$ (with $z_2 \neq 0$), $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$.
• Equation of a Circle: The equation $x^2 + y^2 = r^2$ represents a circle centered at the origin with radius $r$. Equivalently, $|z| = r$ represents a circle of radius $r$ centered at the origin in the complex plane.

Step-by-Step Solution

• Step 1: Express the complex number and identify the goal

  We are given the complex number $w = \frac{\alpha + i}{\alpha - i}$, where $\alpha \in \mathbb{R}$. Our goal is to find the locus of $w$ in the complex plane. We will first use the modulus property to find $|w|$.

• Step 2: Apply the modulus property for division

  We use the property that the modulus of a quotient is the quotient of the moduli: $|w| = \left| \frac{\alpha + i}{\alpha - i} \right| = \frac{|\alpha + i|}{|\alpha - i|}$ This simplifies the calculation because we can now find the moduli of the numerator and denominator separately.

• Step 3: Calculate the modulus of the numerator

  Let $z_1 = \alpha + i$. Then, the modulus of the numerator is: $|z_1| = |\alpha + i| = \sqrt{\alpha^2 + 1^2} = \sqrt{\alpha^2 + 1}$ This is found directly using the definition of the modulus of a complex number.

• Step 4: Calculate the modulus of the denominator

  Let $z_2 = \alpha - i$. Then, the modulus of the denominator is: $|z_2| = |\alpha - i| = \sqrt{\alpha^2 + (-1)^2} = \sqrt{\alpha^2 + 1}$ Notice that the denominator is the complex conjugate of the numerator, and the modulus of a complex number and its conjugate are equal.

• Step 5: Substitute and simplify

  Substitute the calculated moduli back into the expression for $|w|$: $|w| = \frac{\sqrt{\alpha^2 + 1}}{\sqrt{\alpha^2 + 1}} = 1$ Since the numerator and denominator are the same (and non-zero), the result is 1. This shows that the modulus of $w$ is constant and independent of $\alpha$.

• Step 6: Interpret the result geometrically

  Since $|w| = 1$, every point $w$ in the set $S$ is at a distance of 1 from the origin in the complex plane. This means that the locus of $w$ is a circle centered at the origin with a radius of 1.

Common Mistakes & Tips

• Recognizing the $Z/\bar{Z}$ form: The expression $w = \frac{\alpha + i}{\alpha - i}$ is of the form $Z/\bar{Z}$, where $Z = \alpha + i$. This immediately implies that $|w| = 1$, as $|Z| = |\bar{Z}|$.
• Algebraic Errors: When using the Cartesian form method (not used predominantly here but mentioned in the original prompt), be careful with algebraic manipulations, especially when squaring expressions and dealing with $i^2 = -1$.
• Checking for Exclusions: Always consider if any values of $\alpha$ would make the denominator zero. In this case, $\alpha - i$ is never zero for real $\alpha$, so there are no exclusions.

Summary

The modulus calculation shows that $|w| = 1$ for all $\alpha \in \mathbb{R}$. This means that the locus of all such points $w$ is a circle centered at the origin $(0,0)$ with a radius of 1. Therefore, all the points in the set $S$ lie on a circle whose radius is 1.

The final answer is $\boxed{C}$, which corresponds to option (C).