Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + iy$, the modulus is $|z| = \sqrt{x^2 + y^2}$. Also, $|z|^2 = z\bar{z}$, where $\bar{z} = x - iy$ is the complex conjugate of $z$.
• Properties of Modulus: $\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|}$ and $|z_1z_2| = |z_1||z_2|$.
• Equation of a Circle in Complex Form: The general equation of a circle in the complex plane is $|z - z_0| = r$, where $z_0$ is the center of the circle and $r$ is the radius. In Cartesian form, the equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$.

Step-by-Step Solution

Step 1: Rewrite the given equation

We are given $\left| \frac{z - 2i}{z + i} \right| = 2$. We want to manipulate this equation to find the locus of $z$. We can rewrite the equation as $|z - 2i| = 2|z + i|$. Why this step? This step isolates the modulus terms, making it easier to work with and eventually eliminate the modulus signs through squaring.

Step 2: Square both sides

To remove the modulus, square both sides of the equation: $|z - 2i|^2 = (2|z + i|)^2$ $|z - 2i|^2 = 4|z + i|^2$ Why this step? Squaring both sides eliminates the square root inherent in the modulus definition, simplifying the algebraic manipulations.

Step 3: Substitute $z = x + iy$ and expand

Substitute $z = x + iy$ into the equation: $|x + iy - 2i|^2 = 4|x + iy + i|^2$ $|x + i(y - 2)|^2 = 4|x + i(y + 1)|^2$ Using the definition of the modulus, $|a + ib|^2 = a^2 + b^2$, we get: $x^2 + (y - 2)^2 = 4[x^2 + (y + 1)^2]$ Expanding the terms: $x^2 + y^2 - 4y + 4 = 4(x^2 + y^2 + 2y + 1)$ $x^2 + y^2 - 4y + 4 = 4x^2 + 4y^2 + 8y + 4$ Why this step? This step converts the complex equation into a Cartesian equation, which is easier to recognize as a circle equation.

Step 4: Simplify the equation

Rearrange the terms to get a standard form of a circle equation: $0 = 3x^2 + 3y^2 + 12y$ Divide by 3: $x^2 + y^2 + 4y = 0$ Why this step? This step simplifies the equation and prepares it for completing the square.

Step 5: Complete the square

Complete the square for the $y$ terms: $x^2 + (y^2 + 4y + 4) = 4$ $x^2 + (y + 2)^2 = 2^2$ Why this step? Completing the square allows us to identify the center and radius of the circle directly from the equation.

Step 6: Identify the center and radius

The equation is now in the form $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. In this case, $h = 0$, $k = -2$, and $r = 2$. Thus, the center of the circle is $(0, -2)$ and the radius is 2. Why this step? This is the final step where we extract the required information (center and radius) from the equation.

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs, especially when expanding and completing the square.
• Modulus Definition: Ensure you correctly apply the definition of the modulus, both in complex and Cartesian forms.
• Completing the Square: Practice completing the square to avoid errors in identifying the center and radius.

Summary

By substituting $z = x + iy$ into the given equation, squaring both sides, and simplifying, we obtained the equation of a circle in Cartesian form. Completing the square allowed us to identify the center as $(0, -2)$ and the radius as 2. Therefore, $z$ lies on a circle of radius 2 and center $(0, -2)$.

Final Answer

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