Two circles in the complex plane are
`{:(C_(1) : |z-i|=2),(C_(2) : |z-1-2i|=4):}` then

Options

A. `C_(1) and C_(2)` touch each other
B. `C_(1) and C_(2)` intersect at two distinct points
C. `C_(1)` lies within `C_(2)`
D. `C_(2)` lies within `C_(1)`

Answer

To determine the relationship between the two circles given in the complex plane, we will analyze their centers and radii step by step.

### Step 1: Identify the centers and radii of the circles

The equations of the circles are given as:
1. \( C_1: |z - i| = 2 \)
2. \( C_2: |z - (1 + 2i)| = 4 \)

From these equations, we can identify:
- The center of circle \( C_1 \) is at \( i \) (which is \( 0 + 1i \) or \( (0, 1) \) in Cartesian coordinates) and the radius is \( 2 \).
- The center of circle \( C_2 \) is at \( 1 + 2i \) (which is \( (1, 2) \) in Cartesian coordinates) and the radius is \( 4 \).

### Step 2: Calculate the distance between the centers of the circles

To find the distance \( d \) between the centers of the circles \( C_1 \) and \( C_2 \), we use the distance formula:
\[ d = |(0 + 1i) - (1 + 2i)| = |(-1) + (-1)i| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Compare the distance with the sum and difference of the radii

The radii of the circles are:
- Radius of \( C_1 \) = \( 2 \)
- Radius of \( C_2 \) = \( 4 \)

Now, we need to check two conditions:
1. The sum of the radii: \( R_1 + R_2 = 2 + 4 = 6 \)
2. The absolute difference of the radii: \( |R_2 - R_1| = |4 - 2| = 2 \)

### Step 4: Determine the relationship between the circles

Now we compare the distance \( d \) with the sum and difference of the radii:
- \( d = \sqrt{2} \)
- \( R_1 + R_2 = 6 \)
- \( |R_2 - R_1| = 2 \)

Since \( d < |R_2 - R_1| \) (i.e., \( \sqrt{2} < 2 \)), this indicates that circle \( C_1 \) lies entirely within circle \( C_2 \).

### Conclusion

Thus, we conclude that circle \( C_1 \) lies entirely within circle \( C_2 \).