Complex Numbers

Convert to polar form: Then, Evaluate using calculator or De Moivre’s theorem. Result is neither matching any of the options.

Let , then Given , Write Equating: So total = 2003 values of with one extra when , total = 2004 real (a, b) pairs.

Given , take reciprocal: Now since Since , but more importantly: We get Thus numerator becomes zero, and so:

Let , then Now, So, is conjugate of .

By direct computation, we expand the terms based on the binomial expansion and simplify to find that the correct value for the fraction is when is an integer.

The equation does not represent a standard geometric figure like a circle, parabola, or ellipse.
Thus, the answer is "None of these."

Step 1: Let the complex number be represented as , where and are real numbers. Step 2: Substitute into the given equation . Recall that . The equation becomes: Step 3: Group the real and imaginary parts on the right side of the equation. Step 4: Equate the real and imaginary parts on both sides. The left side, , is a real number. Therefore, its imaginary part is 0. Equating the imaginary parts: Equating the real parts: Step 5: Substitute the value into the equation for the real parts. Step 6: Solve the equation for . Before squaring both sides, note that , so we must have , which means . Square both sides: Subtract from both sides: Step 7: Verify the condition . Since , and , the value of is valid. Step 8: Write the solution for and compare with options. The solution is . This matches option (A).