The locus of point z satisfying Re`(z^(2))=0`, is

Options

A. a pair of straight lines
B. a circle
C. a rectangular hyperbola
D. none of these

Answer

To find the locus of the point \( z \) satisfying \( \text{Re}(z^2) = 0 \), we can follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of \( z \).

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

### Step 3: Identify the real part of \( z^2 \)
The real part of \( z^2 \) is given by:
\[ \text{Re}(z^2) = x^2 - y^2 \]

### Step 4: Set the real part equal to zero
According to the problem, we need to satisfy:
\[ \text{Re}(z^2) = 0 \implies x^2 - y^2 = 0 \]

### Step 5: Factor the equation
We can factor the equation:
\[ x^2 - y^2 = (x - y)(x + y) = 0 \]

### Step 6: Solve the factored equations
From the factored form, we have two equations:
1. \( x - y = 0 \) which simplifies to \( y = x \)
2. \( x + y = 0 \) which simplifies to \( y = -x \)

### Step 7: Interpret the results
The equations \( y = x \) and \( y = -x \) represent two lines in the Cartesian plane. Therefore, the locus of the point \( z \) satisfying \( \text{Re}(z^2) = 0 \) is the pair of straight lines given by these equations.

### Final Answer
The locus of the point \( z \) satisfying \( \text{Re}(z^2) = 0 \) is the pair of straight lines:
\[ y = x \quad \text{and} \quad y = -x \]

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