The curve represented by `"Im"(z^(2))=`k, where k is a non-zero real number, is

Options

A. a pair of straight line
B. an ellipse
C. a parabola
D. a hyperbola

Answer

To solve the problem, we need to analyze the equation given by the imaginary part of \( z^2 \) where \( z \) is a complex number. Let's break it down step by step.

### Step 1: Define the complex number
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

**Hint:** Remember that \( z \) can be expressed in terms of its real and imaginary parts.

### Step 2: Compute \( z^2 \)
Now, we calculate \( z^2 \):
\[ z^2 = (x + iy)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + 2xyi \]

**Hint:** Use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \) to expand the square.

### Step 3: Identify the imaginary part
The imaginary part of \( z^2 \) is the coefficient of \( i \):
\[ \text{Im}(z^2) = 2xy \]

**Hint:** The imaginary part of a complex number \( a + bi \) is \( b \).

### Step 4: Set the imaginary part equal to \( k \)
According to the problem, we have:
\[ \text{Im}(z^2) = k \]
Thus,
\[ 2xy = k \]

**Hint:** This step involves equating the imaginary part to the given constant \( k \).

### Step 5: Rearranging the equation
We can rearrange the equation to express it in a different form:
\[ xy = \frac{k}{2} \]
Let \( c = \frac{k}{2} \) (where \( c \) is a non-zero real number since \( k \) is non-zero).

**Hint:** By introducing a new variable, it simplifies the expression.

### Step 6: Identify the type of curve
The equation \( xy = c \) represents a hyperbola in the Cartesian plane.

**Hint:** Recall that the standard form of a hyperbola can be expressed as \( xy = k \) where \( k \) is a constant.

### Final Conclusion
Thus, the curve represented by \( \text{Im}(z^2) = k \) where \( k \) is a non-zero real number is a hyperbola.

**Final Answer:** The curve is a hyperbola.