Number of solutions of eq. `|z|=10and|z-3+4i|=5` is:

Answer

To solve the problem of finding the number of solutions for the equations \( |z| = 10 \) and \( |z - 3 + 4i| = 5 \), we can interpret these equations geometrically.

### Step-by-Step Solution:

1. **Understanding the First Equation**:
The equation \( |z| = 10 \) represents a circle in the complex plane centered at the origin (0, 0) with a radius of 10.

\[ \text{Circle 1: } (x - 0)^2 + (y - 0)^2 = 10^2 \implies x^2 + y^2 = 100 \]

2. **Understanding the Second Equation**:
The equation \( |z - 3 + 4i| = 5 \) can be rewritten as \( |(x - 3) + (y + 4)i| = 5 \). This represents another circle in the complex plane, centered at the point (3, -4) with a radius of 5.

\[ \text{Circle 2: } (x - 3)^2 + (y + 4)^2 = 5^2 \implies (x - 3)^2 + (y + 4)^2 = 25 \]

3. **Finding the Centers and Radii**:
- Center of Circle 1: \( (0, 0) \), Radius \( r_1 = 10 \)
- Center of Circle 2: \( (3, -4) \), Radius \( r_2 = 5 \)

4. **Calculating the Distance Between the Centers**:
The distance \( d \) between the centers of the two circles can be calculated using the distance formula:

\[ d = \sqrt{(3 - 0)^2 + (-4 - 0)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

5. **Using the Circle Touching Condition**:
For two circles to touch each other internally, the following condition must hold:

\[ d = r_1 - r_2 \]

Substituting the values we found:

\[ 5 = 10 - 5 \]

This condition is satisfied, indicating that the two circles touch each other internally.

6. **Conclusion**:
Since the two circles touch at exactly one point, there is exactly one solution for the equations \( |z| = 10 \) and \( |z - 3 + 4i| = 5 \).

\[ \text{Number of solutions: } 1 \]