The system of equation `|z+1+i|=sqrt2 and |z|=3}`, (where `i=sqrt-1`) has

Options

A. no solutions
B. one solution
C. two solution
D. none of these

Answer

To solve the system of equations given by \( |z + 1 + i| = \sqrt{2} \) and \( |z| = 3 \), we can interpret these equations geometrically in the complex plane.

### Step-by-Step Solution:

1. **Rewrite the Equations**:
- The first equation \( |z + 1 + i| = \sqrt{2} \) can be rewritten as \( |z - (-1 - i)| = \sqrt{2} \). This represents a circle in the complex plane centered at the point \((-1, -1)\) with a radius of \(\sqrt{2}\).
- The second equation \( |z| = 3 \) represents a circle centered at the origin \((0, 0)\) with a radius of \(3\).

2. **Identify the Centers and Radii**:
- For the first circle:
- Center: \( (-1, -1) \)
- Radius: \( \sqrt{2} \)
- For the second circle:
- Center: \( (0, 0) \)
- Radius: \( 3 \)

3. **Calculate the Distance Between the Centers**:
- The distance \(d\) between the centers of the two circles can be calculated using the distance formula:
\[ d = \sqrt{(-1 - 0)^2 + (-1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \]

4. **Compare the Distance with the Radii**:
- The sum of the radii of the two circles is:
\[ R_1 + R_2 = \sqrt{2} + 3 \]
- The distance between the centers is \(d = \sqrt{2}\).

5. **Determine the Relationship**:
- Since \(d = \sqrt{2}\) is less than \(R_1 + R_2\) (which is greater than \(\sqrt{2}\)), we check if the circles intersect.
- However, we also need to check if the distance \(d\) is greater than the absolute difference of the radii:
\[ |R_1 - R_2| = |3 - \sqrt{2}| \]
- Since \(3 > \sqrt{2}\), we find that \(d < |R_1 - R_2|\).

6. **Conclusion**:
- Since the distance between the centers \(d\) is less than the difference of the radii, the two circles do not intersect. Therefore, there are no solutions to the system of equations.

### Final Answer:
The system of equations has **no solution**.