Points z in the complex plane satisfying `"Re"(z+1)^(2)=|z|^(2)+1` lie on

Options

A. a circle
B. a parabola
C. an ellipse
D. a hyperbola

Explanation

To solve the problem, we need to find the locus of points \( z \) in the complex plane that satisfy the equation:

\[ \text{Re}((z + 1)^2) = |z|^2 + 1 \]

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

### Step 2: Calculate \( z + 1 \)
We have:
\[ z + 1 = (x + 1) + iy \]

### Step 3: Calculate \( (z + 1)^2 \)
Now, we square \( z + 1 \):
\[ (z + 1)^2 = ((x + 1) + iy)^2 = (x + 1)^2 + 2i(x + 1)y - y^2 \]
Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ = (x + 1)^2 - y^2 + 2i(x + 1)y \]

### Step 4: Find the real part of \( (z + 1)^2 \)
The real part of \( (z + 1)^2 \) is:
\[ \text{Re}((z + 1)^2) = (x + 1)^2 - y^2 \]

### Step 5: Calculate \( |z|^2 \)
The magnitude \( |z|^2 \) is given by:
\[ |z|^2 = x^2 + y^2 \]

### Step 6: Set up the equation
Now we substitute these into the original equation:
\[ (x + 1)^2 - y^2 = x^2 + y^2 + 1 \]

### Step 7: Expand and simplify the equation
Expanding the left side:
\[ (x^2 + 2x + 1) - y^2 = x^2 + y^2 + 1 \]
Now, simplifying:
\[ x^2 + 2x + 1 - y^2 = x^2 + y^2 + 1 \]
Subtract \( x^2 + 1 \) from both sides:
\[ 2x - y^2 = y^2 \]
Combine like terms:
\[ 2x = 2y^2 \]
Dividing both sides by 2 gives:
\[ x = y^2 \]

### Step 8: Identify the locus
The equation \( x = y^2 \) represents a parabola that opens to the right.

### Final Answer
The points \( z \) in the complex plane satisfying the given equation lie on a parabola.

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