Question

Let \( z \) be a non-zero complex number such that \[ z = \frac{16}{\bar{z}}. \] Then the locus of \( z \) is:

• A. a straight line
• B. a parabola
• C. an ellipse
• D. any circle of radius 4
• E. a circle with center at the origin

Hint:

In complex numbers, the equation \( |z|^2 = r^2 \) represents a circle with center at the origin and radius \( r \). In this case, \( x^2 + y^2 = 16 \) corresponds to a circle of radius 4.

Answer 1

Answer: (E)

We are given the equation \( z = \frac{16}{\bar{z}} \), where \( z \) is a non-zero complex number and \( \bar{z} \) represents its complex conjugate.
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers, and \( \bar{z} = x - iy \).
Step 1: Multiply both sides of the equation by \( \bar{z} \) to eliminate the denominator: \[ z \cdot \bar{z} = 16 \] Since \( z \cdot \bar{z} = |z|^2 = x^2 + y^2 \), we have: \[ x^2 + y^2 = 16. \] Step 2: The equation \( x^2 + y^2 = 16 \) represents a circle with radius 4 centered at the origin in the complex plane.
Thus, the locus of \( z \) is a circle with center at the origin and radius 4.
Therefore, the correct answer is option (E).