91Ó°ÊÓ

Complex numbers are key players in the complex plane. They consist of a real part and an imaginary part and are usually written in the form \( z = x + yi \), where \( x \) is the real component, \( y \) is the imaginary component, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \). These numbers are plotted on a two-dimensional plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

• The real part of a complex number, \( \operatorname{Re}(z) \), is simply \( x \).
• The imaginary part of the complex number denoted \( \operatorname{Im}(z) \), is \( y \).

The versatility of complex numbers allows them to represent many kinds of relationships and geometric shapes, including lines and curves on the complex plane.

The absolute value or modulus of a complex number \( z = x + yi \) is a measure of its distance from the origin \((0,0)\) on the complex plane. It is denoted by \(|z|\) and calculated using the formula \(|z| = \sqrt{x^2 + y^2}\). This principle extends when determining the distance between two complex numbers, such as \(|z-a|\), where \(a\) is another complex number. This is the distance from \(z\) to \(a\), given by \(|x-a| + yi \)
In the given exercise, \(|z-2|\) translates to the physical distance from the point \( (x, y) \) to \((2,0)\) which formulates as \( \sqrt{(x - 2)^2 + y^2} \). This finding is essential when analyzing relationships between complex numbers geometrically.

A parabola is a unique curve that can appear in the graphical representation of complex numbers. It usually takes a form similar to \(y^2 = 4(x - h)\), which opens either to the right or left depending on the vertex, \(h\). In the context of our earlier discussion, we analyze a specific parabola equation:

• The equation \(y^2 = 4(x - 1)\) results from evaluating the equality \(|z-2| = \operatorname{Re}(z)\).
• This describes a parabola that opens to the right with its vertex at \((1, 0)\).
• It is symmetric about the x-axis—meaning for every point \((x, y)\), there is a corresponding point \((x, -y)\) present.

This geometrical shape vividly illustrates how certain conditions or constraints, such as the given condition in the exercise, shape complex number relationships. Recognizing such a form and plotting it in the complex plane can illuminate insights into how different complex numbers relate to one another.