Exploring the Range of (z^2) Given (|z-5| leq 6)

This article delves into the mathematical problem of determining the minimum and maximum values of (z^2) under the condition (|z-5| leq 6). We will explore the problem from a geometric and algebraic perspective, illustrating each step with clear explanations and examples.

1. Geometric Interpretation of (|z-5| leq 6)

The inequality (|z-5| leq 6) describes a region on the complex plane where the distance from any point (z) to the fixed point 5 (or (5 0i)) is less than or equal to 6. In other words, it defines a closed disk with a center at 5 and a radius of 6. This can be visualized as:

2. Finding the Extremes of (z^2)

To find the minimum and maximum values of (z^2) under the given constraint, we first express (z^2) in a useful form. Let (z x yi), then:

[z^2 (x yi)^2 x^2 - y^2 2xyi]

The problem can be simplified by considering the extreme values of (z) on the boundary of the disk, which are the endpoints of the interval for (z) in the real axis. From the inequality (|z-5| leq 6), we can directly determine the range for (z):

[-11 leq z leq 1]

Given this range, we can now find the minimum and maximum values of (z^2):

2.1 Minimum Value of (z^2)

The minimum value of (z^2) occurs when (z) is at its smallest positive value on the boundary, which is 1:

[z^2 1^2 1] However, since we are looking for the minimum value of (z^2), we should consider the value of (z^2) when (z -11):

[z^2 (-11)^2 121]

Thus, the minimum value of (z^2) given (|z-5| leq 6) is:

[3]

2.2 Maximum Value of (z^2)

The maximum value of (z^2) occurs when (z) is at its largest positive value on the boundary, which is 1:

[z^2 1^2 1] Again, considering the maximum value of (z^2) at the boundary, where (z -11):

[z^2 (-11)^2 121]

Thus, the maximum value of (z^2) given (|z-5| leq 6) is:

[9]

Conclusion:

The minimum value of (z^2) is 3, and the maximum value is 9.

3. Geometric Interpretation of (z^2 r^2) and Distance in Complex Plane

The geometric interpretation of (z^2 r^2) where (r) is the radius of a circle centered at (-2) can be understood through the distance between centers and the radius. The distance between the centers of the two circles is 3 units, and the radius of the disk is 6 units. Hence, the maximum radius of the circle centered at (-2) is:

[9]

And the minimum radius of this circle is:

[3]

Conclusion:

The minimum and maximum values of (z^2) are 3 and 9, respectively.

4. Summary

In conclusion, the problem of finding the minimum and maximum values of (z^2) given (|z-5| leq 6) can be solved using both algebraic and geometric approaches. By understanding the constraints and interpreting them geometrically, we can determine the range of (z^2). This method is crucial for solving similar problems in complex analysis and has implications in various fields including engineering and physics.

Keywords: complex plane, inequality region, geometric interpretation, minimum and maximum value, distance in complex numbers