If `| (z - i)/(z + 2i)| = 1, |z| = 5/2` then the value of `|z + 3i|`

Options

A. `7/2`
B. `sqrt10`
C. `sqrt5`
D. sqrt3`

Explanation

To solve the problem, we need to find the value of \( |z + 3i| \) given that \( \left| \frac{z - i}{z + 2i} \right| = 1 \) and \( |z| = \frac{5}{2} \).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We start with the equation:
\[ \left| \frac{z - i}{z + 2i} \right| = 1 \]
This implies that:
\[ |z - i| = |z + 2i| \]
This means the distances from the point \( z \) to the points \( i \) and \( -2i \) are equal.

2. **Setting Up the Equation**:
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part. Then we can rewrite the distances:
\[ |z - i| = |(x + iy) - i| = |x + i(y - 1)| = \sqrt{x^2 + (y - 1)^2} \]
\[ |z + 2i| = |(x + iy) + 2i| = |x + i(y + 2)| = \sqrt{x^2 + (y + 2)^2} \]
Setting these equal gives:
\[ \sqrt{x^2 + (y - 1)^2} = \sqrt{x^2 + (y + 2)^2} \]

3. **Squaring Both Sides**:
Squaring both sides eliminates the square roots:
\[ x^2 + (y - 1)^2 = x^2 + (y + 2)^2 \]
Simplifying this, we get:
\[ (y - 1)^2 = (y + 2)^2 \]

4. **Expanding and Simplifying**:
Expanding both sides:
\[ y^2 - 2y + 1 = y^2 + 4y + 4 \]
Canceling \( y^2 \) from both sides:
\[ -2y + 1 = 4y + 4 \]
Rearranging gives:
\[ 6y = -3 \quad \Rightarrow \quad y = -\frac{1}{2} \]

5. **Using the Modulus Condition**:
We know \( |z| = \frac{5}{2} \):
\[ |z| = \sqrt{x^2 + y^2} = \frac{5}{2} \]
Squaring both sides:
\[ x^2 + y^2 = \frac{25}{4} \]
Substituting \( y = -\frac{1}{2} \):
\[ x^2 + \left(-\frac{1}{2}\right)^2 = \frac{25}{4} \]
This simplifies to:
\[ x^2 + \frac{1}{4} = \frac{25}{4} \]
Thus:
\[ x^2 = \frac{25}{4} - \frac{1}{4} = \frac{24}{4} = 6 \]
Therefore, \( x = \pm \sqrt{6} \).

6. **Finding \( |z + 3i| \)**:
Now we need to calculate \( |z + 3i| \):
\[ |z + 3i| = |x + i(y + 3)| = \sqrt{x^2 + (y + 3)^2} \]
Substituting \( y = -\frac{1}{2} \):
\[ |z + 3i| = \sqrt{x^2 + \left(-\frac{1}{2} + 3\right)^2} = \sqrt{x^2 + \left(\frac{5}{2}\right)^2} \]
Thus:
\[ |z + 3i| = \sqrt{x^2 + \frac{25}{4}} \]
Since \( x^2 = 6 \):
\[ |z + 3i| = \sqrt{6 + \frac{25}{4}} = \sqrt{\frac{24}{4} + \frac{25}{4}} = \sqrt{\frac{49}{4}} = \frac{7}{2} \]

### Final Answer:
\[ |z + 3i| = \frac{7}{2} \]