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: {Let the complex number \( z = x + iy \)}
The points corresponding to \( z, \bar{z}, -z, -\bar{z} \) will form a rectangle with vertices \( (x, y), (x, -y), (-x, -y), (-x, y) \). 
Step 2: {Find the area of the rectangle}
Area of rectangle = \( 2x \times 2y = 4xy \). 
Step 3: {Given that the area is \( 2\sqrt{3} \)}
Thus, we have: \[ 4xy = 2\sqrt{3} \Rightarrow 2xy = \sqrt{3}. \] 
Step 4: {Solve for \( x \) and \( y \)}
We know that the rectangle's sides are formed by \( x \) and \( y \). Solving this equation will give us: \[ x = \frac{1}{2}, \quad y = \sqrt{3}. \] Therefore, \( z = \frac{1}{2} + \sqrt{3}i \). 
Step 5: {Verify the answer}
Thus, the correct value for \( z \) is \( \frac{1}{2} + \sqrt{3}i \), which matches option (A). 
 

Answer 2

Step 1: Understand the points forming the rectangle
The points are \( z, \bar{z}, -z, -\bar{z} \). These represent complex numbers and their conjugates and negatives.

Step 2: Visualize the rectangle
- \(z = x + yi\)
- \(\bar{z} = x - yi\)
- \(-z = -x - yi\)
- \(-\bar{z} = -x + yi\)
These four points form a rectangle symmetric about both axes.

Step 3: Calculate side lengths
Length of side along real axis:
\[ |z - \bar{z}| = |(x+yi) - (x - yi)| = |2yi| = 2|y| \]
Length of side along imaginary axis:
\[ |z - (-\bar{z})| = |(x+yi) - (-x + yi)| = |2x| = 2|x| \]

Step 4: Area of rectangle
\[ \text{Area} = \text{side}_1 \times \text{side}_2 = (2|x|) \times (2|y|) = 4|xy| \] Given area = \(2\sqrt{3}\), so
\[ 4|xy| = 2\sqrt{3} \implies |xy| = \frac{\sqrt{3}}{2} \]

Step 5: Find possible values of \(x\) and \(y\)
Try \(x = \frac{1}{2}\) and \(y = \sqrt{3}\), since
\[ \left|\frac{1}{2} \times \sqrt{3}\right| = \frac{\sqrt{3}}{2} \] which satisfies the condition.

Step 6: Write \(z\)
\[ z = \frac{1}{2} + \sqrt{3}i \]

Final Answer: \(\frac{1}{2} + \sqrt{3}i\)