Question

If \( z, \bar{z}, -z, -\bar{z} \) forms a rectangle of area \( 2\sqrt{3} \) square units, then one such \( z \) is:

• A. \( \frac{1}{2} + \sqrt{3}i \)
• B. \( \frac{\sqrt{5} + \sqrt{3}i}{4} \)
• C. \( \frac{3}{2} + \frac{\sqrt{3}i}{2} \)
• D. \( \frac{\sqrt{3} + \sqrt{11}i}{2} \)

Hint:

In problems involving rectangles formed by complex numbers, the vertices of the rectangle are represented by the complex number and its conjugate, as well as their negatives.

Answer 1

The Correct Option is A

Step 1: {Define the complex number \( z = x + iy \)}
The points corresponding to \( z, \bar{z}, -z, -\bar{z} \) form a rectangle with vertices \( (x, y), (x, -y), (-x, -y), (-x, y) \).
Step 2: {Calculate the rectangle's area}
Area of rectangle = \( 2x \times 2y = 4xy \). 
Step 3: {Use the given area value}
Given that the area is \( 2\sqrt{3} \), we have: \[ 4xy = 2\sqrt{3} \Rightarrow 2xy = \sqrt{3}. \] 
Step 4: {Determine \( x \) and \( y \)}
The rectangle's sides are represented by \( x \) and \( y \). Solving the equation \( 2xy = \sqrt{3} \) along with the geometric constraints yields: \[ x = \frac{1}{2}, \quad y = \sqrt{3}. \] Therefore, \( z = \frac{1}{2} + \sqrt{3}i \). 
Step 5: {Confirm the result}
The calculated value for \( z \) is \( \frac{1}{2} + \sqrt{3}i \), which corresponds to option (A).