Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + iy$, the modulus is $|z| = \sqrt{x^2 + y^2}$, representing the distance from the origin to the point $(x, y)$ in the complex plane.
• Geometric Interpretation of $|z - z_0|$: The expression $|z - z_0|$ represents the distance between the complex numbers $z$ and $z_0$ in the complex plane.
• Perpendicular Bisector: The equation $|z - z_1| = |z - z_2|$ represents the locus of points $z$ that are equidistant from $z_1$ and $z_2$, which is the perpendicular bisector of the line segment joining $z_1$ and $z_2$.

Step 1: Use the first condition to find a relationship between x and y

The first condition is $\left| \frac{z - i}{z + 2i} \right| = 1$. This implies $|z - i| = |z + 2i|$. Let $z = x + iy$, where $x$ and $y$ are real numbers. We aim to find a relationship between $x$ and $y$.

$|x + iy - i| = |x + iy + 2i|$ $|x + i(y - 1)| = |x + i(y + 2)|$ $\sqrt{x^2 + (y - 1)^2} = \sqrt{x^2 + (y + 2)^2}$ Squaring both sides, we get $x^2 + (y - 1)^2 = x^2 + (y + 2)^2$ $x^2 + y^2 - 2y + 1 = x^2 + y^2 + 4y + 4$ $-2y + 1 = 4y + 4$ $-3 = 6y$ $y = -\frac{1}{2}$ This tells us that the imaginary part of $z$ is $-\frac{1}{2}$.

Step 2: Use the second condition to find the possible values of x

The second condition is $|z| = \frac{5}{2}$. We have $z = x + iy$ and $y = -\frac{1}{2}$. Substituting into the modulus equation: $|z| = \sqrt{x^2 + y^2} = \frac{5}{2}$ $\sqrt{x^2 + \left(-\frac{1}{2}\right)^2} = \frac{5}{2}$ $x^2 + \frac{1}{4} = \frac{25}{4}$ $x^2 = \frac{24}{4} = 6$ $x = \pm\sqrt{6}$ Therefore, $z = \sqrt{6} - \frac{1}{2}i$ or $z = -\sqrt{6} - \frac{1}{2}i$.

Step 3: Calculate |z + 3i|

We want to find $|z + 3i|$. Let's use $z = \sqrt{6} - \frac{1}{2}i$. $z + 3i = \sqrt{6} - \frac{1}{2}i + 3i = \sqrt{6} + \frac{5}{2}i$ $|z + 3i| = \left|\sqrt{6} + \frac{5}{2}i\right| = \sqrt{(\sqrt{6})^2 + \left(\frac{5}{2}\right)^2} = \sqrt{6 + \frac{25}{4}} = \sqrt{\frac{24}{4} + \frac{25}{4}} = \sqrt{\frac{49}{4}} = \frac{7}{2}$ If we use $z = -\sqrt{6} - \frac{1}{2}i$: $z + 3i = -\sqrt{6} - \frac{1}{2}i + 3i = -\sqrt{6} + \frac{5}{2}i$ $|z + 3i| = \left|-\sqrt{6} + \frac{5}{2}i\right| = \sqrt{(-\sqrt{6})^2 + \left(\frac{5}{2}\right)^2} = \sqrt{6 + \frac{25}{4}} = \sqrt{\frac{24}{4} + \frac{25}{4}} = \sqrt{\frac{49}{4}} = \frac{7}{2}$ In both cases, we get the same result.

Common Mistakes & Tips

• Algebraic Errors: Be very careful with signs and arithmetic, especially when squaring and simplifying expressions.
• Modulus Definition: Remember the correct formula for the modulus of a complex number, $|x + iy| = \sqrt{x^2 + y^2}$.
• Geometric Intuition: Visualizing complex numbers as points in the complex plane can help understand the modulus as a distance.

Summary

We used the given conditions to find the possible values of the complex number $z$. The first condition implied that $z$ lies on the perpendicular bisector of the line segment joining $i$ and $-2i$, which gave us the imaginary part of $z$. The second condition gave us the modulus of $z$, which we used to find the real part of $z$. Finally, we calculated $|z + 3i|$ using either possible value of $z$, obtaining $\frac{7}{2}$.

Final Answer

The final answer is \boxed{7/2}, which corresponds to option (D).