Key Concepts and Formulas

• Modulus of a Complex Number: If $z = x + iy$, where $x$ and $y$ are real numbers, then the modulus of $z$ is $|z| = \sqrt{x^2 + y^2}$. Geometrically, $|z|$ represents the distance of the point $(x, y)$ from the origin in the complex plane.
• Geometric Interpretation of |z - z₀|: $|z - z_0|$ represents the distance between the complex number $z$ and the complex number $z_0$ in the Argand plane.
• Equation of a Circle: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $\sqrt{g^2 + f^2 - c}$. The standard form is $(x-h)^2 + (y-k)^2 = r^2$, with center $(h,k)$ and radius $r$.

Step-by-Step Solution

1. Rewrite the Given Equation We begin by isolating one of the modulus terms. This will make the subsequent algebraic manipulations easier. The given equation is $2|z + 3i| - |z - i| = 0$. We rewrite this as: $2|z + 3i| = |z - i|$

2. Substitute z = x + iy Now, we substitute $z = x + iy$, where $x$ and $y$ are real numbers, to convert the complex number equation into a Cartesian equation. This allows us to work with real variables. $2|x + iy + 3i| = |x + iy - i|$

3. Group Real and Imaginary Parts We group the real and imaginary components within each modulus for clarity. $2|x + i(y + 3)| = |x + i(y - 1)|$

4. Apply the Modulus Definition Using the definition $|a + bi| = \sqrt{a^2 + b^2}$, we express the moduli in terms of $x$ and $y$. $2\sqrt{x^2 + (y + 3)^2} = \sqrt{x^2 + (y - 1)^2}$

5. Square Both Sides To eliminate the square roots, we square both sides of the equation. This simplifies the equation significantly. $\left(2\sqrt{x^2 + (y + 3)^2}\right)^2 = \left(\sqrt{x^2 + (y - 1)^2}\right)^2$ $4(x^2 + (y + 3)^2) = x^2 + (y - 1)^2$

6. Expand and Simplify We expand the squared terms and simplify the equation by collecting like terms. $4(x^2 + y^2 + 6y + 9) = x^2 + y^2 - 2y + 1$ $4x^2 + 4y^2 + 24y + 36 = x^2 + y^2 - 2y + 1$ $3x^2 + 3y^2 + 26y + 35 = 0$

7. Divide by 3 To get the standard form of the circle equation, we divide the entire equation by 3 so that the coefficients of $x^2$ and $y^2$ are both 1. $x^2 + y^2 + \frac{26}{3}y + \frac{35}{3} = 0$

8. Complete the Square (Optional but Recommended) While not strictly needed to find the radius, completing the square helps visualize the center. To complete the square for the $y$ terms, we add and subtract $(\frac{26}{6})^2 = (\frac{13}{3})^2 = \frac{169}{9}$ inside the equation. $x^2 + \left(y^2 + \frac{26}{3}y + \frac{169}{9}\right) - \frac{169}{9} + \frac{35}{3} = 0$ $x^2 + \left(y + \frac{13}{3}\right)^2 = \frac{169}{9} - \frac{35}{3} = \frac{169 - 105}{9} = \frac{64}{9}$ $x^2 + \left(y + \frac{13}{3}\right)^2 = \frac{64}{9}$

9. Identify Center and Radius Comparing to the standard circle equation $(x-h)^2 + (y-k)^2 = r^2$, we have $h=0$, $k=-\frac{13}{3}$, and $r^2 = \frac{64}{9}$. Thus, the radius is: $r = \sqrt{\frac{64}{9}} = \frac{8}{3}$ The center of the circle is $(0, -\frac{13}{3})$.

Common Mistakes & Tips:

• Sign Errors: Be extremely careful with signs when expanding and simplifying equations. A single sign error can lead to an incorrect result.
• Squaring Correctly: When squaring both sides of an equation, make sure to square all terms, including any coefficients.
• Completing the Square: If you're not comfortable with finding the radius directly from the general equation, completing the square is a reliable method, though it takes a little longer.

Summary By substituting $z = x + iy$ into the given equation and simplifying, we transformed the complex number equation into a Cartesian equation representing a circle. Through algebraic manipulation, we determined the radius of this circle to be $\frac{8}{3}$. This matches option (A).

Final Answer The final answer is $\boxed{8/3}$, which corresponds to option (A).