The locus of the point z is the Argand plane for which `|z +1|^(2) + |z-1|^(2)= 4` is a

Options

A. Straight line
B. Pair of straight lines
C. Parabola
D. Circle

Answer

To solve the problem of finding the locus of the point \( z \) in the Argand plane for which \( |z + 1|^2 + |z - 1|^2 = 4 \), we can follow these steps:

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

### Step 2: Rewrite the equation using \( z \)
The equation becomes:
\[ |z + 1|^2 + |z - 1|^2 = 4 \]
Substituting \( z \):
\[ |(x + 1) + iy|^2 + |(x - 1) + iy|^2 = 4 \]

### Step 3: Calculate the modulus squared
Using the formula \( |a + ib|^2 = a^2 + b^2 \):
\[ |(x + 1) + iy|^2 = (x + 1)^2 + y^2 \]
\[ |(x - 1) + iy|^2 = (x - 1)^2 + y^2 \]
Thus, we can rewrite the equation as:
\[ (x + 1)^2 + y^2 + (x - 1)^2 + y^2 = 4 \]

### Step 4: Simplify the equation
Combine the terms:
\[ (x + 1)^2 + (x - 1)^2 + 2y^2 = 4 \]
Expanding the squares:
\[ (x^2 + 2x + 1) + (x^2 - 2x + 1) + 2y^2 = 4 \]
This simplifies to:
\[ 2x^2 + 2 + 2y^2 = 4 \]

### Step 5: Further simplify
Dividing the entire equation by 2:
\[ x^2 + y^2 + 1 = 2 \]
This leads to:
\[ x^2 + y^2 = 1 \]

### Step 6: Identify the locus
The equation \( x^2 + y^2 = 1 \) represents a circle with a radius of 1 centered at the origin.

### Conclusion
Therefore, the locus of the point \( z \) is a circle.

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