Complex Numbers

Step 1: {Given that }
Thus, we know that .
Step 2: {Write the sum of complex numbers}
Now, we have:
Step 3: {Conclusion}
By calculating the sum and using the properties of the magnitudes, we find:
Step 4: {Verify the result}
Thus, the correct value is , which matches option (C).

Step 1: {Let the complex number }
The points corresponding to will form a rectangle with vertices .
Step 2: {Find the area of the rectangle}
Area of rectangle = .
Step 3: {Given that the area is }
Thus, we have:
Step 4: {Solve for and }
We know that the rectangle's sides are formed by and . Solving this equation will give us: Therefore, .
Step 5: {Verify the answer}
Thus, the correct value for is , which matches option (A).

Given the complex number , we need to find the locus such that . This equation states that the distance from to 1 is equal to the distance from to -1.

Let's expand these distances:

and .

Set them equal: .

Square both sides to eliminate the square roots: .

Expand both sides: .

Cancel common terms ( , , ) from both sides: .

Combine like terms: .

Solve for : .

This implies that the complex number lies on the line where , which is the imaginary axis.