Understanding the Geometric Representation of Complex Equations

The question asks for the geometric shape represented by the equation \( \text{Re} (z^2 - i) = 2 \), where \( z \) is a complex number and \( \text{Re} \) denotes the real part of the complex number.

Let the complex number \( z \) be represented as \( z = x + iy \), where \( x \) and \( y \) are real numbers representing the real and imaginary parts, respectively. The geometric representation will be in the Cartesian plane using coordinates \( (x, y) \).

Step-by-Step Analysis of the Equation

1. Substitute \( z = x + iy \) into the expression \( z^2 - i \):

   First, calculate \( z^2 \):

   \( z^2 = (x + iy)^2 \)

   Expanding this, we get:

   \( z^2 = x^2 + 2(x)(iy) + (iy)^2 \)

   \( z^2 = x^2 + 2ixy + i^2 y^2 \)

   Since \( i^2 = -1 \), this becomes:

   \( z^2 = x^2 + 2ixy - y^2 \)

   \( z^2 = (x^2 - y^2) + i(2xy) \)

   Now, subtract \( i \) from \( z^2 \):

   \( z^2 - i = (x^2 - y^2) + i(2xy) - i \)

   \( z^2 - i = (x^2 - y^2) + i(2xy - 1) \)

2. Find the Real Part:

   The expression \( z^2 - i \) is in the form \( A + iB \), where \( A = x^2 - y^2 \) is the real part and \( B = 2xy - 1 \) is the imaginary part.

   The equation given is \( \text{Re} (z^2 - i) = 2 \).

   Therefore, we take the real part of \( z^2 - i \) and set it equal to 2:

   \( \text{Re} (z^2 - i) = x^2 - y^2 \)

   So, the equation in terms of \( x \) and \( y \) is:

   \( x^2 - y^2 = 2 \)

3. Identify the Geometric Shape:

   The equation \( x^2 - y^2 = 2 \) is in the standard form of a hyperbola centered at the origin:

   \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \)

   Our equation \( x^2 - y^2 = 2 \) can be rewritten as:

   \( \frac{x^2}{2} - \frac{y^2}{2} = 1 \)

   Comparing this to the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), we see that \( a^2 = 2 \) and \( b^2 = 2 \). This means \( a = \sqrt{2} \) and \( b = \sqrt{2} \).

4. Characteristics of the Shape:

   The equation \( \frac{x^2}{2} - \frac{y^2}{2} = 1 \) represents a hyperbola with \( a = b \).

   A hyperbola where the lengths of the semi-major and semi-minor axes are equal (\( a = b \)) is called a rectangular hyperbola (or equilateral hyperbola). For such a hyperbola, the asymptotes are perpendicular to each other.

Conclusion on Geometric Representation

The equation \( \text{Re} (z^2 - i) = 2 \) transforms into \( x^2 - y^2 = 2 \) in the Cartesian plane, which is the equation of a rectangular hyperbola.

Revision Table: Conic Sections Overview

Additional Information: Complex Numbers and Geometry

Complex numbers provide a powerful way to represent points in a 2D plane (the complex plane or Argand plane). An equation involving a complex variable \( z \) often translates into a geometric shape in this plane.

• The form \( z = x + iy \) is the connection between complex numbers and Cartesian coordinates \( (x, y) \).
• Operations on \( z \), like squaring \( z^2 \) or taking the real part \( \text{Re}(z) \), correspond to transformations or conditions that define loci (geometric shapes) in the \( x-y \) plane.
• Understanding how to convert a complex equation into an equation involving \( x \) and \( y \) is crucial for identifying the geometric shape it represents. This involves substituting \( z = x + iy \) and separating the real and imaginary parts.

In this specific problem, evaluating \( z^2 - i \) and then taking the real part allowed us to translate the complex equation \( \text{Re} (z^2 - i) = 2 \) into the familiar Cartesian equation \( x^2 - y^2 = 2 \), which is a rectangular hyperbola.