91Ó°ÊÓ

The modulus of a complex number is a fundamental concept. It measures how far a point, representing the complex number, is from the origin in the complex plane. For a complex number expressed as \( Z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part, the modulus of \( Z \) is calculated using the formula:

• \( |Z| = \sqrt{x^2 + y^2} \)

This modulus represents the "size" or "magnitude" of the complex number. In our specific exercise, the given condition involves the modulus, which signifies a specific distance constraint between two points on the complex plane: \( Z \) and \( \frac{4}{Z} \). The provided constraint \( |Z - \frac{4}{Z}| = 2 \) ensures that the distance between these points is exactly 2 units, serving as the key to solving the problem.

Understanding distance in the complex plane aids in visualizing and solving problems involving complex numbers. The complex plane is similar to a Cartesian coordinate system, with the real and imaginary parts of a complex number acting like the x and y coordinates. The distance between two complex numbers \( Z_1 = x_1 + y_1i \) and \( Z_2 = x_2 + y_2i \) is calculated using:

• \( |Z_1 - Z_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

In the specific problem, the expression \( |Z - \frac{4}{Z}| = 2 \) represents the Euclidean distance formula applied in the complex plane. This formula becomes a guide in transitioning from abstract complex representations to genuine geometric interpretations, allowing us to translate the algebraic condition into a comprehensible geometric visualization. This visualization forms the foundation in assessing potential choices for the modulus \(|Z|\).

Quadratic equations often appear in problems involving complex numbers due to their squared terms. In the exercise, after substituting and simplifying, we encountered a quadratic equation of the form \( u^2 - 8u + 16 = 0 \). Here, \( u \) represents \( r^2 \), which comes from expressing the modulus \(|Z|\) in terms of real numbers.To solve quadratic equations:

• Utilize the general formula for roots: \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• In this case, the equation \((u - 4)^2 = 0\) was solved by recognizing its perfect square form, directly providing \( u = 4 \).

The intersection of algebraic manipulation and root-solving skills leads us to determine that the maximum value of \(|Z|\) is 2. This value satisfies the quadratic equation and the initial condition given, demonstrating that mastering quadratic solutions is key to unlocking distance and modulus constraints in complex number problems.