If |z-i| ≤ 2 and`z_0 = 5+3i`, the maximum value of `|i z + z_0|` is

Options

A. `5+sqrt13`
B. `5+sqrt2`
C. 7
D. 8

Answer

To solve the problem, we need to find the maximum value of \( |iz + z_0| \) given that \( |z - i| \leq 2 \) and \( z_0 = 5 + 3i \).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
The condition \( |z - i| \leq 2 \) describes a circle in the complex plane centered at \( i \) (which is \( 0 + 1i \)) with a radius of 2. This means that \( z \) can take any value within or on the boundary of this circle.

2. **Expressing \( |iz + z_0| \)**:
We need to find \( |iz + z_0| \). First, substitute \( z_0 \):
\[ z_0 = 5 + 3i \]
So, we have:
\[ |iz + (5 + 3i)| = |iz + 5 + 3i| \]

3. **Rearranging the Expression**:
We can rearrange the expression:
\[ |iz + 5 + 3i| = |iz + (5 + 3i)| \]
We can factor out \( i \):
\[ = |i(z + 5i)| \]
The modulus of a product is the product of the moduli:
\[ = |i| \cdot |z + 5i| = 1 \cdot |z + 5i| = |z + 5i| \]

4. **Finding the Maximum of \( |z + 5i| \)**:
Now, we need to maximize \( |z + 5i| \). Since \( z \) can be expressed as \( z = x + yi \) (where \( x \) and \( y \) are real numbers), we have:
\[ |z + 5i| = |x + (y + 5)i| = \sqrt{x^2 + (y + 5)^2} \]

5. **Using the Circle Condition**:
From the condition \( |z - i| \leq 2 \), we can express \( z \) as:
\[ z = x + (y + 1)i \quad \text{where } \sqrt{x^2 + (y - 1)^2} \leq 2 \]
This means that \( (x, y - 1) \) lies within or on the boundary of a circle centered at \( (0, 0) \) with radius 2.

6. **Finding the Center and Radius**:
The center of the circle is at \( (0, 1) \) and the radius is 2. The maximum distance from the center \( (0, 1) \) to the point \( (0, 5) \) (which corresponds to \( z + 5i \)) is:
\[ \text{Distance} = |(0, 5) - (0, 1)| = 4 \]

7. **Calculating Maximum Value**:
Therefore, the maximum value of \( |z + 5i| \) occurs when \( z \) is at the point on the circle that is farthest from \( -5i \):
\[ \text{Maximum value} = \text{Radius} + \text{Distance from center to } -5i = 2 + 4 = 6 \]

### Final Answer:
Thus, the maximum value of \( |iz + z_0| \) is \( 7 \).