If `|z-i| lt 1,` then the value of `|z + 12-6i|` is less than

Options

A. 14
B. 2
C. 28
D. None of these

Answer

To solve the problem, we need to analyze the given condition and find the required value step by step.

### Step 1: Understand the given condition
We are given that \( |z - i| < 1 \). This means that the complex number \( z \) lies within a circle in the complex plane centered at \( 0 + i \) (which corresponds to the point \( (0, 1) \) in the Cartesian plane) with a radius of \( 1 \).

### Step 2: Identify the center and radius of the circle
The center of the circle is \( (0, 1) \) and the radius is \( 1 \). Therefore, the region defined by \( |z - i| < 1 \) includes all points \( z \) that are less than \( 1 \) unit away from the point \( (0, 1) \).

### Step 3: Find the distance from the center to the point \((-12, -6)\)
Next, we need to find the distance \( d \) from the center of the circle \( (0, 1) \) to the point \( (-12, -6) \). The distance formula in the Cartesian plane is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(-12 - 0)^2 + (-6 - 1)^2} = \sqrt{(-12)^2 + (-7)^2} = \sqrt{144 + 49} = \sqrt{193} \]

### Step 4: Determine the minimum and maximum distances
Now we can find the minimum and maximum distances from the circle to the point \((-12, -6)\):
- The minimum distance from the circle to the point is \( d - r \), where \( r \) is the radius of the circle:
\[ \text{Minimum distance} = d - 1 = \sqrt{193} - 1 \]
- The maximum distance from the circle to the point is \( d + r \):
\[ \text{Maximum distance} = d + 1 = \sqrt{193} + 1 \]

### Step 5: Establish the inequality for \( |z + 12 - 6i| \)
From the distances calculated, we can write:
\[ \sqrt{193} - 1 < |z + 12 - 6i| < \sqrt{193} + 1 \]

### Step 6: Find the upper bound
Since we are asked for the value of \( |z + 12 - 6i| \) to be less than a certain value, we focus on the upper bound:
\[ |z + 12 - 6i| < \sqrt{193} + 1 \]

### Step 7: Approximate \( \sqrt{193} \)
Calculating \( \sqrt{193} \):
\[ \sqrt{193} \approx 13.89 \quad (\text{since } 13^2 = 169 \text{ and } 14^2 = 196) \]
Thus,
\[ \sqrt{193} + 1 \approx 14.89 \]

### Conclusion
Therefore, the value of \( |z + 12 - 6i| \) is less than approximately \( 14.89 \). Since the question asks for a value less than this, we can conclude that the answer is:
\[ \text{The value of } |z + 12 - 6i| \text{ is less than } 14. \]