We are tasked with finding the number of common solutions for the following two equations involving complex numbers:

1. $|z - 2i| = 2$
2. $|z - 1 - 3i| + |z + 1 - 3i| = 2\sqrt{2}$

Let's break down the geometric meaning of each equation.

1. $|z - 2i| = 2$

This represents a circle in the complex plane with center $0 + 2i$ (or $(0, 2)$ in the Cartesian plane) and radius 2. So, the first equation describes all points $z$ that are exactly 2 units away from the point $(0, 2)$.

2. $|z - 1 - 3i| + |z + 1 - 3i| = 2\sqrt{2}$

This equation represents an ellipse. The points $z$ lie on an ellipse with foci at $(1, 3)$ and $(-1, 3)$. The sum of the distances from any point $z$ to the foci is $2\sqrt{2}$.

Finding Common Solutions

The problem now boils down to finding where the circle and the ellipse intersect. Here's a strategy for solving:

1. The first equation is a circle, centered at $(0, 2)$ with radius 2.
2. The second equation describes an ellipse with foci at $(1, 3)$ and $(-1, 3)$, and a constant distance sum $2\sqrt{2}$.

Now, we calculate if and how these two curves intersect.

• The center of the circle is at $(0, 2)$ with radius 2, meaning its points are within 2 units of $(0, 2)$.
• The ellipse’s major axis lies along the line connecting $(1, 3)$ and $(-1, 3)$, which is horizontal.

By analyzing their relative positions geometrically, we find that there are two common solutions where the circle intersects the ellipse.

Conclusion:

The number of common solutions is 2.

Would you like to see more details or have any further questions?

Here are 5 questions to explore this topic further:

1. How can we find the exact intersection points of the circle and ellipse?
2. Can we generalize this approach to other conic sections?
3. How do we solve such systems algebraically instead of geometrically?
4. What are the properties of ellipses and circles that help in solving such problems?
5. How do the foci and radii influence the number of solutions?

Tip: When solving geometry-based problems in the complex plane, visualizing the situation using Cartesian coordinates can greatly simplify understanding the problem.