For a complex number Z, if `|Z-i|le2 and Z_(1)=5+3i`, then the maximum value of `|iZ+Z_(1)|` is (where, `i^(2)=-1`)

Options

A. `7+sqrt(13)`
B. `7+sqrt(12)`
C. 7
D. `sqrt(34)-2`

Answer

To solve the problem, we need to find the maximum value of \( |iZ + Z_1| \) given the conditions \( |Z - i| \leq 2 \) and \( Z_1 = 5 + 3i \).

### Step-by-step Solution:

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

2. **Expressing the Target**:
We want to find the maximum value of \( |iZ + Z_1| \). We can rewrite this as:
\[ |iZ + (5 + 3i)| = |iZ + 5 + 3i| = |5 + (iZ + 3i)| \]
This can be simplified to:
\[ |5 + i(Z + 3)| \]

3. **Substituting \( Z \)**:
Let \( Z = x + yi \), where \( x \) and \( y \) are real numbers. Then:
\[ iZ = i(x + yi) = -y + xi \]
Thus,
\[ iZ + Z_1 = (-y + 5) + (x + 3)i \]

4. **Finding the Magnitude**:
The magnitude is given by:
\[ |(-y + 5) + (x + 3)i| = \sqrt{(-y + 5)^2 + (x + 3)^2} \]

5. **Using the Circle Condition**:
Since \( |Z - i| \leq 2 \), we have:
\[ |(x + (y - 1)i)| \leq 2 \]
This implies:
\[ x^2 + (y - 1)^2 \leq 4 \]
This describes the same circle in the complex plane.

6. **Maximizing the Expression**:
To maximize \( |iZ + Z_1| \), we can use the triangle inequality:
\[ |iZ + Z_1| \leq |iZ| + |Z_1| \]
Here, \( |Z_1| = |5 + 3i| = \sqrt{5^2 + 3^2} = \sqrt{34} \).

7. **Finding the Maximum Value of \( |iZ| \)**:
The maximum value of \( |iZ| \) occurs when \( |Z| \) is maximized. The maximum distance from the center of the circle (which is at \( (0, 1) \)) to the origin (which is \( (0, 0) \)) is:
\[ \text{Maximum } |Z| = \text{Radius} + \text{Distance from center to origin} = 2 + 1 = 3 \]

8. **Final Calculation**:
Therefore, the maximum value of \( |iZ| \) is 3. Thus:
\[ |iZ + Z_1| \leq 3 + \sqrt{34} \]

### Conclusion:
The maximum value of \( |iZ + Z_1| \) is \( 3 + \sqrt{34} \).