It seems you're asking about the expression \( |z - i| = \sqrt{2} \). This represents a complex number problem involving distances in the complex plane.

### Understanding the Problem:

The expression \( |z - i| = \sqrt{2} \) can be interpreted geometrically as follows:

- \( z \) is a complex number that we can represent as \( z = x + iy \), where \( x \) and \( y \) are real numbers (i.e., \( z = x + iy \in \mathbb{C} \)).
- \( |z - i| \) represents the distance between the complex number \( z \) and \( i \) in the complex plane. The number \( i \) corresponds to the point \( (0, 1) \) in the complex plane.

### Geometrical Interpretation:
The equation \( |z - i| = \sqrt{2} \) describes a circle in the complex plane with a center at \( i \) (which is \( (0,1) \)) and a radius of \( \sqrt{2} \).

### Detailed Solution:

1. Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.
   
2. The distance between \( z \) and \( i \) is given by:

   \[
   |z - i| = |(x + iy) - i| = |x + i(y - 1)|
   \]

3. By definition of modulus of a complex number, we have:

   \[
   |x + i(y - 1)| = \sqrt{x^2 + (y - 1)^2}
   \]

4. The equation \( |z - i| = \sqrt{2} \) becomes:

   \[
   \sqrt{x^2 + (y - 1)^2} = \sqrt{2}
   \]

5. Squaring both sides:

   \[
   x^2 + (y - 1)^2 = 2
   \]

This is the equation of a circle with center \( (0,1) \) and radius \( \sqrt{2} \) in the complex plane.

### Summary:
The equation \( |z - i| = \sqrt{2} \) represents a circle in the complex plane, centered at \( (0,1) \) with a radius of \( \sqrt{2} \).

Would you like further details or have any other questions?

### Related Questions:
1. What is the geometric interpretation of a complex modulus in general?
2. How do you solve for specific points \( z \) that satisfy \( |z - i| = \sqrt{2} \)?
3. What other conic sections can be represented using complex numbers?
4. How can this problem be extended to three-dimensional space using quaternions?
5. What is the relationship between the modulus and the argument of a complex number?

**Tip**: Visualizing complex numbers on the complex plane can help you understand relationships like distances and rotations more clearly.

Explore the geometric interpretation of the complex modulus equation |z - i| = \sqrt{2}, which represents a circle in the complex plane. Learn how to break down the equation and solve for the center and radius, and understand the connection to complex numbers. Suitable for Grades 10-12 students.

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