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

Options

A. `sqrt(10)`
B. `(7)/(2)`
C. `(15)/(4)`
D. `2sqrt(3)`

Answer

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**:
The condition \( \left| \frac{z - i}{z + 2i} \right| = 1 \) implies that the magnitudes of the numerator and the denominator are equal:
\[ |z - i| = |z + 2i| \]

2. **Expressing \( z \) in terms of real and imaginary parts**:
Let \( z = a + bi \), where \( a \) and \( b \) are real numbers. Then we can rewrite the magnitudes:
\[ |z - i| = |(a + bi) - i| = |a + (b - 1)i| = \sqrt{a^2 + (b - 1)^2} \]
\[ |z + 2i| = |(a + bi) + 2i| = |a + (b + 2)i| = \sqrt{a^2 + (b + 2)^2} \]

3. **Setting up the equation**:
From the equality of the magnitudes, we have:
\[ \sqrt{a^2 + (b - 1)^2} = \sqrt{a^2 + (b + 2)^2} \]
Squaring both sides yields:
\[ a^2 + (b - 1)^2 = a^2 + (b + 2)^2 \]
Simplifying gives:
\[ (b - 1)^2 = (b + 2)^2 \]

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

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

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

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