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 start with the given conditions:

1. \(|\frac{z - i}{z + 2i}| = 1\)
2. \(|z| = \frac{5}{2}\)

We need to find the value of \(|z + 3i|\).

### Step 1: Analyze the first condition

From the first condition, we have:

\[ |\frac{z - i}{z + 2i}| = 1 \]

This implies that:

\[ |z - i| = |z + 2i| \]

### Step 2: Express \(z\) in terms of its real and imaginary parts

Let \(z = a + bi\), where \(a\) and \(b\) are real numbers. Then we can write:

\[ |z - i| = |(a + bi) - i| = |a + (b - 1)i| = \sqrt{a^2 + (b - 1)^2} \]
\[ |z + 2i| = |(a + bi) + 2i| = |a + (b + 2)i| = \sqrt{a^2 + (b + 2)^2} \]

### Step 3: Set the magnitudes equal

Setting the two magnitudes equal gives us:

\[ \sqrt{a^2 + (b - 1)^2} = \sqrt{a^2 + (b + 2)^2} \]

### Step 4: Square both sides

Squaring both sides, we get:

\[ a^2 + (b - 1)^2 = a^2 + (b + 2)^2 \]

This simplifies to:

\[ (b - 1)^2 = (b + 2)^2 \]

### Step 5: Expand and simplify

Expanding both sides:

\[ b^2 - 2b + 1 = b^2 + 4b + 4 \]

Subtract \(b^2\) from both sides:

\[ -2b + 1 = 4b + 4 \]

Rearranging gives:

\[ -2b - 4b = 4 - 1 \]
\[ -6b = 3 \implies b = -\frac{1}{2} \]

### Step 6: Use the second condition

Now we use the second condition \(|z| = \frac{5}{2}\):

\[ |z| = \sqrt{a^2 + b^2} = \frac{5}{2} \]

Substituting \(b = -\frac{1}{2}\):

\[ \sqrt{a^2 + \left(-\frac{1}{2}\right)^2} = \frac{5}{2} \]
\[ \sqrt{a^2 + \frac{1}{4}} = \frac{5}{2} \]

### Step 7: Square both sides again

Squaring both sides gives:

\[ a^2 + \frac{1}{4} = \left(\frac{5}{2}\right)^2 \]
\[ a^2 + \frac{1}{4} = \frac{25}{4} \]

### Step 8: Solve for \(a^2\)

Subtract \(\frac{1}{4}\) from both sides:

\[ a^2 = \frac{25}{4} - \frac{1}{4} = \frac{24}{4} = 6 \]

Thus, \(a = \sqrt{6}\) or \(a = -\sqrt{6}\).

### Step 9: Find \(|z + 3i|\)

Now we can find \(|z + 3i|\):

\[ z + 3i = a + \left(-\frac{1}{2} + 3\right)i = a + \frac{5}{2}i \]

Calculating the modulus:

\[ |z + 3i| = \sqrt{a^2 + \left(\frac{5}{2}\right)^2} \]
\[ = \sqrt{6 + \frac{25}{4}} = \sqrt{\frac{24}{4} + \frac{25}{4}} = \sqrt{\frac{49}{4}} = \frac{7}{2} \]

### Final Answer

Thus, the value of \(|z + 3i|\) is:

\[ \frac{7}{2} \]