Geometrically `Re(z^(2)-i)=2,where i=sqrt(-1)` and Re is the real part, represents.

Options

A. circle
B. ell ipse
C. rectangular hyperbola
D. parabola

Explanation

To solve the problem geometrically, we need to analyze the expression given: \( \text{Re}(z^2 - i) = 2 \), where \( i = \sqrt{-1} \).

### Step-by-Step Solution:

1. **Express \( z \) in terms of real and imaginary parts**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

2. **Calculate \( z^2 \)**:
\[ z^2 = (x + iy)^2 = x^2 + 2xyi - y^2 = x^2 - y^2 + 2xyi \]

3. **Subtract \( i \) from \( z^2 \)**:
\[ z^2 - i = (x^2 - y^2 + 2xyi) - i = x^2 - y^2 + (2xy - 1)i \]

4. **Identify the real part**:
The real part of \( z^2 - i \) is:
\[ \text{Re}(z^2 - i) = x^2 - y^2 \]

5. **Set the real part equal to 2**:
According to the problem, we have:
\[ x^2 - y^2 = 2 \]

6. **Recognize the equation**:
The equation \( x^2 - y^2 = 2 \) represents a rectangular hyperbola in the Cartesian coordinate system.

### Conclusion:
Thus, geometrically, the equation \( \text{Re}(z^2 - i) = 2 \) represents a rectangular hyperbola.