If the complex number z is such that `|z-1|le1 and |z-2|=1` find the maximum possible value `|z|^(2)`

Answer

To solve the problem, we need to analyze the given conditions for the complex number \( z \) and find the maximum possible value of \( |z|^2 \).

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- The first condition is \( |z - 1| \leq 1 \). This represents the interior and boundary of a circle centered at \( (1, 0) \) with a radius of \( 1 \).
- The second condition is \( |z - 2| = 1 \). This represents the boundary of a circle centered at \( (2, 0) \) with a radius of \( 1 \).

2. **Graphing the Circles**:
- The first circle (from the first condition) has its center at \( (1, 0) \) and includes all points within and on the circle. It extends from \( (0, 0) \) to \( (2, 0) \) along the x-axis.
- The second circle (from the second condition) has its center at \( (2, 0) \) and includes only the points on the circumference, which extends from \( (1, 0) \) to \( (3, 0) \).

3. **Finding the Intersection**:
- The points satisfying both conditions will lie on the arc of the second circle that intersects with the first circle. The intersection points occur where the circles touch or overlap.

4. **Identifying Key Points**:
- The point \( A \) on the first circle where it intersects the x-axis is \( (2, 0) \).
- The point \( B \) on the second circle where it intersects the x-axis is \( (3, 0) \).

5. **Calculating Distances**:
- The distance from the origin \( O(0, 0) \) to point \( A(2, 0) \) is \( OA = 2 \).
- The distance from the origin \( O(0, 0) \) to point \( B(3, 0) \) is \( OB = 3 \).

6. **Using the Pythagorean Theorem**:
- The maximum distance \( OC \) from the origin to the arc can be calculated using the Pythagorean theorem. The maximum distance occurs at the point where the two circles are tangent to each other.
- We can set up the equation:
\[ OC^2 = OB^2 - BT^2 \]
where \( BT \) is the vertical distance from the center of the second circle to the x-axis, which is \( 1 \).
- Thus, we have:
\[ OC^2 = 2^2 - 1^2 = 4 - 1 = 3 \]

7. **Finding the Maximum Value of \( |z|^2 \)**:
- Therefore, the maximum value of \( |z|^2 \) is \( OC^2 = 3 \).

### Final Answer:
The maximum possible value of \( |z|^2 \) is \( 3 \).

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